SEMFGROUPS HAVING QUASl-FROBENI’US ALGEBRAS Thesis £00 “in Doqnc 0‘ D“. D. HECHIGAN STATE UNIVERSITY Ronald R. Wen-get 1965 “ ___.;.:§§. ‘ l Michigan State 3 m University rl This is to certify that the thesis entitled SERVIIEDLOUPS HAVING QUAS I—FROEEII IUS ALGEERAS presented by Ronald H . Wen ger has been accepted towards fulfillment of the requirements for Ph.D. degree in Matherfiabi CS Major professor I Date 7/2?/ A? /7é£: 0-169 ABSTRACT SEMIGROUPS HAVING QUASI—FROBENIUS ALGEBRAS by'Ronald.H.'Ienger Finite semigroups having semisimple algebras over fields have been characterized by Munn [7]. This characterization involves a restriction concerning the characteristic of the field and cer-- tain subgroups of the semigroup. The purpose here is to determine which finite semigroups have quasi-Frobenius algebras. The main result of section 2 is that this restriction on the field can be dropped and the semigroup algebra remains quasi-Frobenius. Theorem: Let F be a field of characteristic c. If a semigroup S has a principal series in which each factor is c—nonsingular over F, then F(S) is quasi-Frobenius. This theorem implies all finite inverse semigroups have quasi- Frobenius algebras over arbitrary fields. The remaining sections are devoted to the investigation of semigroups S which have a principal series 5 - SOD SIDH‘DSrtl’ in which each principal factor Si/Sitl’ is null, a group, or a group with zero. A semigroup S is defined to be of tzpe_9 if S is a group or S satisfies the conditions: 1) S is commutative with zero, 2) S has a principal series S = SOZDSiZLH°ZDSr*l=(O),'with 8-8 a group, and $1 nilpotent, l 5) 8638055 for each 36 S. Ronald H. Wenger The key result in this case is: Theorem: Let S be a commutative semigroup and F a field. F(S) is quasi-Frobenius if and only if S has an ideal series $403112." Dqu, with Ii/Iifl a semigroup of type c and with F(Ii/I ) quasi-Frobenius for each i = 0,1,...,q. i+l This theorem is used to reduce the commutative case to the simpler case of semigroups of type C. The group algebra F(G) is a subalgebra of F(S) for a type C semigroup. F(G) is Frobenius and this is used to find conditions on F(S). The primitive idempotents of F(S) are found to be in F(G). The result: If e1 and e2 are primitive idempotents of F(G)QF(S), then elF(G) 9" e2F(G) as right F(G)- modules if and only if elF(S) g e2F(S) as right F(S)-modules,is proved. These results can be used to prove that if a semigroup S is of type C, then F(S) is Frobenius if F(S) is quasi-Frobenius. The remaining sections are used to examine the question of when semigroups of type C have quasi-Frobenius algebras. This problem is reduced to that for semigroups of type C with one element in each J-class and these are said to be of type D. The most useful theorem then is: If T is a semigroup of type D with a principal series T a ToDT 3...3Tr+l= (0), then F(T) is quasi-Frobenius if and only if 1 r(F(T1)l = F(Tr>- Ronald H. wenger This result is used to extend the class of semigroups with quasi-Frobenius algebras. In the last section, a construction of some semigroups of type C from semigroups of type D and abelian groups is given. This construction is used to form semigroups of type C with quasi- Frobenius algebras from semigroups of type D which are known to have quasi-Frobenius algebras. SEMIGROUPS HAVING QUASI-FROBENIUS ALGEBRAS By\ Ronald Hi ‘Ienger A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1965 s '_1 C ‘ r-u~ J . \4, -_‘ K ACKNOILEDGEMENTS The author wishes to thank Professor'w. E. Deskins for his help with this thesis. This work was supported, in part, by the National Science Foundation. ii To Sherry and my Parents iii TABLE 95 CONTENTS Introduction ......... . ............ . ...................... 1. Preliminaries .. ...................... . ............... 2. lunn algebras ....... ........................... . ..... 5. A reduction .. ...... . ........ .... ......... . . ...... . 4. Semigroups of types C, C', and D and their algrbras .. 5. A construction for some semigroups of type C from semigroups of type D ................................. 60 Appendix 0 o 000000000000000 0 oooooooooo o 0000000000 o o o o 0 Bibliography .. ...... .............. ............ . ....... . . iv 16 27 47 59 65 TABLE g1: CONTENTS Page Introduction ......... . ................................... l 1. Preliminaries ....................................... 4 2. lunn algebras ....... ........................... . ..... 9 S. A reduction ................. ...... ......... . ........ . 16 4. Semigroups of types C, C', and D and their algrbras .. 27 5. A construction for some semigroups of type C from semigroups of type D ............................... .. 47 6. Appendix .. ......................... . .......... ..... 59 . ....... . . 65 Bibliography .. ...... .......... ................ iv ‘fix‘r" INTRODUCTION Semigroups S with semisimple algebras F(S) over a field F have been characterized by Munn [7]. The goal here is to extend some of these results and to find others for semigroups for which F(S) may not be semisimple. As the theory of non-semisimple algebras in general is not very complete, a natural question is "For what semigroups S is F(S) quasi-Frobenius or Frobenius?" Quasi-Frobenius and Frobenius algebras have been studied rather extensively by'lakayama [8,9] and others so some reasonable characterizations for them are known. Another reason for asking this question is that some of the structure theory for semigroups is related to their subgroups and groups have Frobenius algebras over arbitrary fields. The following results are obtained: In section 2, Munn's result for semisimple algebras is general- ized. His result involves the restriction that the characteristic of the field does not divide the orders of certain groups associated.with the semigroups. The main result of this section drops this condition and provides a large class of quasi-Frobenius semigroup algebras. Theorem: Let F be a field of characteristic c. If a semigroup S has a principal series in which each factor is c-nonsingular over F, then F(S) is quasi-Frobenius. This theorem is used to prove that finite inverse semigroups have quasi-Frobenius algebras over arbitrary fields. Such semigroups are being extensively studied by Munn and others. Then, in particular, a semigroup S with a principal series S=Sb2381:)-~:38r*1 in which each 51/81,.1 is a group with zero, or a group‘has a quasi-Frobenius algebra over an arbitrary field. The question then is raised “What happens if some of the Si/S1+1 are null semigroups?” The prototype for the semigroup for which each principal factor is null, a group with zero, or a group is the abelian semigroup. Thus, in subsequent sections, semigroups for which each principal factor is either null, a group, or a group with zero are examined. The fundamental theorem for the whole investigation is: Theorem: Let S be a commutative semigroup and F a field. F(S) is quasi-Frobenius if and only if S has an ideal series SalIoDIlD'HDIm-l’ with Ii/Ii+l a semigroup of type C and with F(Ii/I.*1) quasi-Frobenius for each i=0,l,...,q. 1 Definition: A semigroup S is of type C' if S is a group or 5 satisfies the following conditions: 1) S has a zero, 2) S has a principal series 5330:)sl;)'°£35r¢1=(0) with 5-81 a group G, and S1 nilpotent, 5) se 53085 for each se 5. S is said to be of type_§ if S is commutative and of type C'. S is of type D if it is of type C and has precisely one element in each J-class. This theorem reduces the problem to that for type-C semigroups. All of the remaining sections are devoted to studying semigroups of types C, C', and D, with the theorem above in mind. For if one can tell when semigroups of type C have quasi—Frobenius algebras then the theorem can be used to answer the question for all finite commutative semigroups. Semigroups of type C with quasi-Frobenius algebras are not com— pletely characterizedbut the following results are obtained. Theorem: If S is of type C" and F(S) is quasi-Frobenius then F(S) is Frobenius. The problem for semigroups S of type C with group G can be related to that for type D semigroups when the characteristic of F does not divide the order of C. It is found that the set of J—classes for 8 forms a semigroup, say T, of type D. This leads to the investigation of semigroups of type D. The most useful theorem then is: Theorem: Let T be a semigroup of type D with a principal series =(O). Then F(T) is quasi-Frobenius if and only T=To 3T 3 . 0 .3T 1 r+l if r(F(Tl))=F(Tr). (Here r(F(T1))=(aeF(T):F(T1)a=(0)). This theorem is used to prove various necessary and some sufficient conditions for a semigroup of type D to have a quasiéFrobenius algebra. Section 4 shows how some semigroups of type C are conStruCted from semigroups of type D and abelian groups. This leads to other semi- groups of type C with quasi-Frobenius algebras. The only semigroups considered here are finite semigroups. 1. Preliminaries Recent books will be cited as references whenever possible. All of the remarks about semigroups can be found in [2]. Notation: If B is a subset of A, then A-B 8 (26 A: a dB). If F is a field, c(F) is the characteristic of F. If I is a set, c(X) represents the cardinality of 1. ¢ ‘will denote the empty set. Semigrggps: A semigroup is a nonempty set S with one binary, associative operation. If a semigroup S does not have an identity, one can be adjoined. Let SU(1) - S1 with 101 - l and loa = a for each as S. A subset I of S is a left ideal of S if SI§:I, a right ideal of S if IS§:I, and a two-sided ideal of S if it is both a left and a right ideal of S. By "ideal" is meant a two-sided ideal. If S has a minimal ideal M, i.e., an ideal which does not properly contain an ideal of S, then M is unique and is called the kernel of S. If S has a zero, O, then I = (O). A semigroup S with a zero is called nilpotent if Sn= (0) for some positive integer n. S is called gel} if 52¢ (00. A semigroup 8 With a zero is said to be O-simple if 32% (O) and S pr0perly contains no nonzero ideal. S is called gimple if S properly contains no two-sided ideals. An element a of S is called regular if aeaSa. S is said to be regular if each of its elements is regular. Two elements a and b of S are inverses of one another if aba = a and bab = b. S is M if there is an a€S such that each element of S is a power of a. A cgggruence relation R on a semigroup S is an equivalence relation on S with the additional property that if a,b€ S and aRb then (ca)R(cb) and (ac)R(bc) for each c in S. If [a] - (be S: bRa), the composition [a][b] - [ab] gives ([a]: aeS) the structure of a semigroup which will be denoted by S/R. If I is an ideal of 5, define the relation R on S as aRb if and only if a e b or both a and b are in I. R is a congruence and the semigroup S/R is written S/I, and is called the Rees factor semigrogp 2;; _S_ modulo I. Because of its construction, 3/1 is often written as (S—IlLJI or (S-I)LJz, where z is the zero of 5/1 and products are defined as: ab - 2 if abe I, and ab is the product in S otherwise. This identification is sometimes convenient. The following theorem is often used. Theorem: Let I be an ideal of S, and let u be the natural homo- morphism of 5 onto S/I. Then a induces a one-to-one, inclusion- preserving mapping A-oAn - A/l, of the set of all ideals A of 5 containing I upon the set of all ideals of 8/1, and (S/I)/(A/I) 3 S/A. If a is in S, the set (be S: SlaSl= Slel) = Ja is called the J-class of S corresponding to a. (Note that if as SarlaS for each as S, then SlaS:L - $85. If S is commutative, then be Ja if and only if bSl= aSl.) If aeS, SlaSl is called the mncipal ideal generated by _a_. 1(a) = SlaSluJa is an ideal of S and SlaSI/I(a) is called a principal factor of S. For convenience, if S is a semigroup and $713 the empty set, let S/¢ mean S. .An ideal series for a semigroup S is a chain (1) s - soaslaé-«Dsrflzd , with 51 an ideal of s, i = o, 1;", r+1. An ideal series is called a principal series if there is no ideal of S strictly between Si and 81.1, i = O,l,°",r+l. If (I) is an ideal series for S, the semigroups Si/Si+l’ i = O,l,-°°,r+l, are called the factors of the series. The following theorems are used. Theorem: Let S be a semigroup admitting a principal series (I). Then the factors of (l) are isomorphic in some order to the principal factors of S. In particular, any two principal series of 5 have isomorphic factors. The last term in any principal series of S is the kernel. The connection between the factors si/Si+l of a principal series for S and the principal factors can be written as Si/Si*1“ SlaSl/I(a) if ae S.-S -S 1 1+1, Si 1+1' series if and only if Sins1+1 is a J-class for each i. Ja. Thus, an ideal series (1) is a principal Theorem: Each principal factor of any semigroup S is Opsimple, simple, or null. Only if S has a kernel is there a simple principal factor. Note that all finite semigroups have principal series and that any ideal series can be refined to a principal series for such semigroups. Let S be a finite semigroup and let F be a field. If X is a subset of S let X' = X—(O). If S has no zero or O #:x, then X' = X. ) In particular, let S' = S-(O). Let F(S) = ( 2 ass: ase F), and 59 8' define, for 2 as and 2 b,s' in F(S), s‘S' S s'€S' S 2 ass . 2 bs,s' if and only if as- b8 for each 555'. 865' 5'6 5' Define also, 2 a s 4- 2 bs,s' =3 2 (as4v bs)s, and s€S' S s'GS' aes' (‘ 2 ass)( 5.: bs,s') = 2 ( 2 asbs,)t. ses' 5'6 5' t€S' ss'at Iith these definitions F(S) is a finite dimensional, associative algebra with the elements of S' as a basis. 5 can be considered a subset of F(S) if one identi fies it with the set (l-s: 3:58) (1 the identity of F), with the zero of F identified with the zero of S if S has a zero. F(S) is called the semigroup algebra (contracted) of 8 over F. Theorem: If T is an ideal of S, then F(T) is an ideal of F(S) and F(S)/F(T) 9' F(S/T) as algebras. If X is a subset of S, then F(X) will denote the linear subspace (S ax: a er) of F(S). xeX' x x Algbras: The following remarks are found in [3] unless otherwise indicated. All algebras considered are finite dimensional, associative algebras over a field F. I If I is a subset of an algebra A, then r(X) = (at A: 13 = (0)) is called the right annihilator of X in A. In like manner, the set 1(1) = (as A: ax=(o)) is called the left annihilator of x. Definition: An algebra A with identity over a field F is quasi- Frobenius if a) r(l(R)) = R and l(r(L)) = L for each right ideal R and each left ideal L of A. A is Frobenius if, in addition. b) (r(L):F) + (L:F) = (A:F) and (l(R):F) + (R:F) = (A:F) for all left ideals L and right ideals R of A, where (X2F) means the dimension of the linear space X over F. Nakayama proved [8, Theorem 3, p. 619] that not all left and right ideals need be checked. Theorem: Let N be the radical of A. If a) holds for every nilpotent simple left ideal L and every nilpotent simple right ideal R of A, as well as for the zero ideal and N itself, then A is quasi- Frobenius. If, moreover, b)holds for such ideals then A is Frobenius. The following remarks can be made about quasi-Frobenius and Frobenius algebras, A. If A = B + C is an algebra direct sum, then A is quasi-Frobenius (Frobenius) if and only if both B and C are quasi—Frobenius (Frobenius). Group algebras are Frobenius. Definition: An idempotent f in an algebra A is called primitive if f cannot be written as the sum of two orthogonal idempotents. I.e., 2, r: = fi, i = 1,2, flfz a rzrl a o. ‘If f is a primitive idempotent of A then Af is called a (left) fifff principal indecomposable module. If N is the radical of A, then every irreducible left A-module is isomorphic to Af/Nf for some primitive idempotent f of A. These modules are very important when discussing non-semisimple (or semisimple) algebras. 2. Eggg algebras The following theorem is proved in this section: Theorem: Let F be a field of characteristic 0. If a semigroup S has a principal series in which each factor is a c-nonsingular semigroup over F, then F(S) is quasi-Frobenius. This result provides a large class of semigroups with quasi- Frobenius semigroup algebras. It is a partial generalization of a theorem by Munn [7, Theorem 6.4, pp. 10], in which semigroups with semisimple algebras are characterized. It will be used to prove that finite inverse semigroups have quasiéFrobenius algebras over arbitrary fields. Such semigroups have been studied extensively by Munn and others because they have properties similar to those of groups. This theorem is proved by induction on the length of a principal series for S. This technique is used quite often and its success depends on the following lemma. Lemma 2.1: Let B be an ideal in the algebra A. If B and A/B are quasi-Frobenius (Frobenius), then A is quasi-Frobenius (Frobenius). nggi: Let e be the identity of B. Then A - B+C, an algebra direct sum, where C e (ce.A: ce = ec = O) [1, Theorem 23, pp. 30]. But both B and C 5 A/B are quasi-Frobenius (Frobenius) by hypothesis so A is quasi—Frobenius (Frobenius) as their direct sum, by a remark in the preliminaries. Many semigroup algebras can be related to total matric algebras over group algebras. The next result is useful in such cases. If A is an algebra, let Mn(A) denote the algebra of nxn matrices over A. 10 Lemma 2.2: Mn(A) is quasi-Frobenius if and only if A is quasi- Frobenius. Egmagk: In this proof and elsewhere, if X is a subset of an algebra, then r(X) will mean the right annihilator of X in the appropriate algebra. The same remark holds for left annihilators. For example, if X is a subset of an algebra A and Mn(X) is the subset of MniA) consisting of all matrices with entries from I, then r(fln(X)) means the right annihilator of Mh(X) in un(t), while un(r(x)) means the set of all nxn matrices with entries from r(X), the right annihilator of X in A. Pzggg: The fact that if A is quasi-Frobenius then.Mn(A) is also, is the content of [9, lemma 2, p. 11]. Conversely, suppose Mn(A) is quasi-Frobenius. If L is a left ideal of A such that l(r(L)) f L, consider Mn(L). Clearly r(Mn(L))2Mn(r(L)). Suppose Be r(Mn(L)). Then YB = O for each I in Mn(L). Let (x)ki denote the nxn matrix with x as its (k,i) entry and zeros elsewhere. If bij / O is the (i,j) entry of B then T ’ o _ (X)kiB - xbil xb12 - . . xbin = (O) for each x in L. _ ° l Then Xbij‘ O for each x in L so bij is in r(L). Thus Be Mn(r(L)) so that r(MnlIJ) = Muir(L)). Dually, l(r(Mn(L))) = Mn(l(r(L))). Thus l(r(L)) ,1 L implies l(r(Mn(L))) Nina). This proves l(r(L)) = L for each left ideal L of A. Similarly, r(l(R)) = R for each right ideal R of A, so A is quasi-Frobenius. 11 Lemma 2.2 has also been proved by the author for Frobenius algebras but the proof is a bit tedious and is unnecessary for the following development so it is not included here. The semigroup algebras which are encountered are of a more general form than Mn(A). Let A be an algebra and let M(A;m,n,;P) be the algebra obtained from the set of mxn matrices with the usual addition but with multiplication defined as 8-0 - BPC, where P is an nxm matrix over A. Definition 2.1: M(A;m,n;P) is called the Mumn algebra over I with sandwich matrix P. [2, p. 162] In general, if an algebra A has an identity e, then an element a is said to be nonsingular if there is an element b in A such that ab = e; then also be = e [1, p. 13] Munn proved [7, Theorem 4.7, p. 7] that M(A3m,n3P) is semisimple if and only if P is nonsingular in Mh(A) and A is semisimple. In the process of proving this he actually proves [2, Iemma.5.18, p. 165]: M(A:m,n;P) has an identity if and only if A has an identity and P is nonsingular. (Then P.lis the identity of M(A;m,n;P).) The result M(A;n,n;P) 5'M(A;n,n;Uh) ‘ Mn(A) if P is nonsingular and A has an identity e, is established and used in the proof of this theorem [7, 4.8, p.8], where Un is the nxn matrix with e's on the main diagonal and zeros elsewhere. These relationships can be used to prove lemma 2.3: M(A;m,n;P) is quasi-Frobenius if and only if i) A is quasi—Frobenius and ii) P is nonsingular in Mn(A). 12 If M(A;m,n;P) is quasi-Frobenius then it has an identity, Proof: namely P. . By the remarks above H(A;n,n;P) gun“). Then Mn(A) is Conversely, if A quasi-Frobenius so A is quasi-Frobenius by lemma 2.2. is quasi-Frobenius, then Hn(A) is quasi—Frobenius by lemma 2.2. If P is nonsingular, then M(A;n,n;P) chin“) is quasi-Frobenius. Conditions under which P is nonsingular are given in [7] and in [2]. The semigroups which are of interest here are constructed as Let G be a group and Co = GlJ(O). Go is a semigroup if follows. Go is called g-O = O-g’= O, 02' O, and other products are those in G. Let m and n be positive integers, and let P be an a group with zero. nxm matrix with entries from GO. Let M°(G;m,n;P) be the set of all mxn matrices with entries from G0 with at most one nonzero entry. (a;i,q) ‘will represent the element of M°(G;m,n;P)'with a as the entry in the (i,q) position and zeros elsewhere, as CO. Identify the elements (O;i,q) with o for all i and q. Let P = (pqi). Then u°(G;m,n;P) is a semigroup if products are defined as (a;i,q)(b;j,r) 8 (aqubgi,r). Definition 2.2: M°(G;m,n;P) is the Bees matrix semigroup over g: ‘with sandwich matrix P. [2, p.87] The same construction can be carried out with G rather than Go and then the semigroup is denoted by'M(G;m,n;P). The following things are known about such semigroups. MO(G;m,n;P) is regular if and only if each row and each column of P contains a nonzero entry [2, Lemma 3.1, p. 89]. A finite semigroup is Obsimple (simple) if and only if it is isomorphic with a regular Rees matrix semigroup over a group with zero (without zero). 15 Thus, whenever a O—simple semigroup arises, one may assume that it has a Rees matrix representation. This will be done. It is clear by the construction of M°(G;m,n:P) that its contracted semigroup algebra over a field F is M(F(G);m,n:P), a Munn algebra over a group algebra. Then as a corollary to lemma 2.5, onehas: Corollary 1: A O-simple (simple) semigroup S has a quasi-Frobenius algebra over a field F if and only if S 3'M°(G3n,n:P) (S gIM(G;n,n:P)) with P nonsingular in.Mn(F(G)). “ The corollary follows since group algebras are quasi-Frobenius. Definition 2.3: A O—simple semigroup S will be called c-nonsingular if S =.MO(G:n,n;P), where P is nonsingular in Mn(F(G)) and c(F) - c. [7, p. 9] A corresponding definition is given for simple semigroups. Munn’s main result [7, Theorem 6.4, p. LC] is: Theorem: Let F(S) be the algebra of 8 over a field F with c(F)=c. Then F(S) is semisimple if and only if each of the principal factors of S is a c-nonsingular semigroup, and c, if nonzero, does not divide the orders of any of the corresponding groups. This can be extended. Theorem 2.1: Let c(F) = c. If a semigroup S has a principal series in which each factor is c-nonsingular over F then F(S) is quasi- Frobenius. Proof: Let S a 50:)sl;)uc)Sr+l be a principal series for S, ‘with Sr+l = (0) if S has a zero and Sr¢1 empty otherwise. Assume sr+l- (O) in the proof for it is the same when Sr+l is empty. If r = 0, then S = SO:)Sl= (O) and S = 50 is 0-simple and c-nonsingular 14 over F so F(S) is quasi-Frobenius by corollary 1. Suppose r 2:1 and assume that the result is true for all semigroups of this type with a principal series of length less than r+l. Then F(Sl) is quasi-Frobenius and F(S)/F(Sl) “.F(S/Sl) is quasi-Frobenius since 5/61 is O-simple and c-nonsingular. Then F(S) is quasi-Frobenius by lemma 2.1. The converse of this theorem - F(S) quasi-Frobenius implies each principal factor of S is c-nonsingular - is false as examples and further investigation will show. The following remark can be made, however. (0). Corollary 1: Let S have a principal series 5 3 $035130-'35r,1‘ If F(Si) has an identity for i a O,1,...,r, then Si/Siwl is c-nonSingular for i - 0,1,. .,r, and F(S) is quasi-Frobenius. Proof: If F(Si) has an identity then F(Si)/F(Si+l) g F(Si/Sifil) so F(Si/si+l) could Suppose ‘. 2 has an identity. If 51/514,1 is null then 819:8“,1 is O-simple (or simple). ) 9‘ M(F(G);m,n;P) and this liunn Thus not have an identity. Then Si/Si+1 a 0 . . Si/Si+l M (G,m,n,.P). Then Mei/Si,1 algebra has an identity if and only if m a n and P is nonsingular. Si/S1+1 is c-nonsingular for each i. Then F(S) is quasi-Frobenius by theorem 2.1. Another corollary to theorem 2.1 can be stated after these definitions. A semigroup S is a Brandt semigrogp if Definition 2.4: s §2M°(c;n,n;uh). [2, Theorem 5.9, p. 102] A semigroup S is called an inverse semigroup Definition 2.5: if each element has a unique inverse. [2, p. 28] 15 In [7], Mann proves that each principal factor of a finite inverse semigroup is a Brandt semigroup. Corollary 2: A finite inverse semigroup S has a quasi—Frobenius algebra over any field F. be a principal series for S. Proof: Let S = 3951395171 Then Si/S1+1 is a Brandt semigroup for i - O,l,...,r. Then si/Si+l is isomorphic to IO(G;n,n;Uh) by definition 2.4:(M°(G:n,n;Uh) depends on i.). But the identity matrix Uh is nonsingular over F(G) for each field F so Si/Sid is a c-nonsingular semigroup for each i 8 O,l,...,r. Then by the theorem F(S) is quasi-Frobenius. By the same argument, any semigroup for which each principal factor is a group with zero (or a group) has a quasi-Frobenius algebra over Section 3 deals with a generalization of such a semigroup. every field. There, all non-null principal factors are assumed to be groups with or without a zero. 3. A reduction The main result of this section is: Theorem: Let S be a commutative semigroup. If F(S) is quasi- Frobenius then S has a principal series with components Ik/ k . O,l,...,q, such that F(Ik/lk+l) is quasi-Frobenius for each Conversely, if S has such a principal series and log S, Ik+1’ k : O,l,...,q. then F(S) is quasi-Frobenius. This theorem will reduce the study of commutative semigroups with This quasi-Frobenius algebras to the study of semigroups of type C. reduction is important because the structure of a semigroup S of type C is influenced by a certain subgroup G of S. Then the fact that F(G) is quasi-Frobenius over arbitrary fields can be used in studying The problem is more complicated than might be expected as a F(S). semigroup, unlike a group, can have a quasi-Frobenius algebra over a field of one characteristic but fail to have such an algebra over a field of a different characteristic. The other difficulty is that, as opposed to semisimple algebras, subalgebras and difference algebras of quasi-Frobenius algebras need not be quasi-Frobenius. The results here are not generalizations of those in the preceding Here semigroups with null principal factors are permitted section. but the non-null factors are required to be groups (with or without zero). The prototype for this case is the commutative semigroup. A few remarks can be made about an arbitrary finite semigroup S for which F(S) has an identity over some field F. 16 17 Lemma 3.1: If the semigroup algebra F(S) has an identity e, then 56 SsnsS for each s in S. Proof: Consider S as embedded in F(S). Then in F(S), es = se = s for each s in S. If e - 2 att, ate F, then es - 2 atts = s te S' t6 5' implies tZS'attS - s = 0 so, by the linear independence of the 6 elements of 5' over F, ts = s for some t in S. In like manner, there is a t' in S such that st' = s, since se 8 3. Thus 3658088 for each s in S. Lemma 3.2: If S = SODSID~DSN1 is a principal series for S and F(S) has an identity e, then S/S:L is a c-nonsingular semigroup, c(F) - c. M: F(S) has an identity implies F(S/Sl) 9‘ F(S)/F(Sl) has an identity. A principal factor 5/51 is null, O-simple or simple (see . 2 ' the preliminaries). If S/S is null S ;S . If e = 2 a t , then 1 1 t6 5' t e = 2 att + 2 att and, for as S—Sl, ' te 55--Sl te 51 s = es s 2 a ts + 2 atts is in F(Sl) as F(Sl) is an ideal of te S-S ta 8' l l F(S) and $2; 51’ This contradicts 56 8-51. Then 8/51 is 0-simple (or simple). Then, as observed in the preceding section, F(S/Sl) has an identity if and only if 3/81 is a c—nonsingular semigroup. In particular, these lemmas apply when F(S) is quasi-Frobenius. Lemma 3.3: If S is a commutative O—simple (simple) semigroup, then S is an abelian group with zero (without zero). l8 .2322£: If S is a O—simple (simple) semigroup, then it has a Rees matrix representation, say S 9" M°(G;m,n;P). The elements of such a semigroup can be written as (a;i,q) as in section 2. Then (aii,q)(b;:).r) = (aqub;1.r) and (b;3.r)(a;i,q) = (bpriaijm), where p is the (q,j) entry of P and pri is the (r,i) entry. If S is qj commutative, then m = n = l and G must be abelian. Thus, S g M°(G;l,l; (p)) (pf 0 since S is non-null). Then M°(G;l,l,(p)) Go under the mapping (ap-1;1,1)-—9a, a in Co, since (p‘l;l,l) is the identity of M°(G;l,l;(p)) and (ap-131,l)(bp-l;l,l) 8 (ap—lpbp-l;l,l) a ((ab)p-l;l,l). Thus S is a group with zero. If S is commutative with a principal series 5 = SoDle'HDSrfl. (Here sr+1 will always be assumed to be zero if S has a zero and empty otherwise.) then Si/Si+1’ i e O,l,...,r, is either null, an abelian group with zero, or an abelian group. This last case occurs only if i = r and Sr+l is empty. Then Sr is the kernel of S (see the prelim- inaries). If S is commutative and F(S) is quasi-Frobenius then F(S) has an identity and lemma 3.2 implies 5/81 is a group with zero (or without zero). It is convenient to give certain classes of semigroups a special notation. Definition 3.1: A semigroup S is of type C‘ if either S is a group or S satisfies the following conditions: (1) S has a zero, (2) S has a principal series 5 3503513'Hnsr4-13 (O), with 8.51 a group and S1 nilpotent, (3) as $3035 for each as S. 19 S is said to be of $223.9 if S is a commutative semigroup of type C'. S is said to be of pype_D if S is of type C and Si-si+l contains precisely one element, 1 - O,l,...,r. (I.e., S has precisely one element in each J-class.) Lemma 3.4: If S is of type C' and R is a congruence on S, then S/B is of type C'. If S is of type C, then S/R is of type C. M: Let S '- 50:) le"°DSr+1 be a principal series for S. Then S/R ' So/RQSl/RQH-QSNl/R is a principal series for 8/8. S/R-Sl/R ' (S-Sl)/B is a group since the homomorphic image of a group is a group. If S? = (0), then (Sl/B)n=(S§/R) - (O) so Sl/R is nilpotent. To see that S/R has property (3), let as S and write SP. for the image of 5 under R. seSsnsS since S is of type 0'. Let s = ts - st‘, t and t' in S. Then SE = (tR)(sR) = (sR)(t'R)e (S/R)(sR)n(sR)(S/R) for each SE in S/R. Thus S/R is of type C'. A homomorphic image of a commutative semigroup is clearly comm- utative so if S is of type C, than 5/8 is of type C. In what follows, let S be commutative unless otherwise specified. Let 5:80:313'°°Dsr+l be an arbitrary principal series for S with Sr+lg (O) or sr+l empty. Q will denote the subset of the set (Ogl,...,r) such that Si/Si+l is non-null if and only if i is in Q. Suppose Q ' (io,il,...,iq), ik< in if k < n. Let 1k: Sik’ k = O,l,...,q, and, for convenience, let qul= Sr+l° Then $2103 IlD---DIqDIé+1 is an ideal series for S such that Ik/Ik+l is a semigroup which satisfies all the conditions for a semigroup of type C except perhaps (3). 20 Definition 3.2: The factors Ik/Ikfl, k = O,l,...,q, are called These components depend upon the principal series chosen for S. Lemma 3.5 shows that the integer q is an invariant for 8, however. Lemma 3.5: The number of components in a principal series for S is the same for each principal series of S. Pppp£: In the preliminaries a theorem was stated which said that given any two principal series for S, there is a one-to-one corres- pondence between their factors so that corresponding factors are isom- orphic. Thus, any two principal series for S have the same number of non-null factors, say q+l. Then each principal series for S has q+1 components. The following example illustrates the fact that the components for different principal series need not be isomorphic. Let S have the multiplication table: e a b 0 e e a b O a a. a O O b b O O O O O O 0 O S has two principal series: (1) SD(a,b,0):D(b,O)D(O), and (2) 53(a,b,o):>(a,o):>(o). In (1), let H03 S, H1= (a,b,O), H2= (0). Then the components are Ho/Hl‘ (e,z) (using the identification described in the preliminaries), and 111/112 giHl a (a,b,O). 21 In (2) let Io= s, Il= (a,o), I2 = (o). The components for this series are Io/Il‘ (e,b,z) and Il/I2= I1g (a,O). (1) has no component isomorphic with (e,b,z) = Io/Il of (2) so the components of (l) and (2) are not isomorphic. Note also that S satisfies property (3), se 55 for each se 5, since S has an identity e. However, Hl/H2g H1. (a,b,0) is a factor for (l) and H1 does not satisfy (3) since b e le= (0). Both components of (2) satisfy (3) as semi- groups, and are therefore of type C. It is surprising that if a commutative semigroup S satisfies condition (3), then S has a principal series the components, Ik/Ik+1’ of which also satisfy it. This series can be used to prove that if F(S) is quasi-Frobenius (Frobenius) then F(Ik/Ik+1) is quasi-Frobenius (Frobenius) for each k, and conversely. This reduces the question to investigating semigroups of type C, since the components Ik/Ik+l for this series are of type C. In the next section it is found that semigroups of type C have quasi-Frobenius algebras only if they have Frobenius algebras. Theorem 3.1: If S is a commutative semigroup having property (3), se 68 for each s in S, then S has a principal series for which each component has this property. Proof: The proof is by induction on q, where q¢l is the number of components of S. Note that property (3) is preserved by homo- morphisms by lemma 3.4. Observe also that property (3) implies that in any pr1nc1pal series S=Sb:381;)w~:)Sr*1 for S, 50/51 is non-null, forctherwise if as 80-51, then s é sS_C_Sl since SZQSI. If q = 0, then S = I0 is the only component of S so the result is true. Assume q 2:1 and suppose the theorem is true for commutative 22 semigroups with less than q¢l components. If S=S°D Sl'DH-DSN1 is a principal series for S (as before Sr+1= (O) or is empty), let p be the maximal non-ne gative integer such that Sp/Sp #1 is non-null. Among all the principal series for S, choose one in which the corres- ponding p is maximal. (I.e., the last non-null factor is "as close to the bottom" of the series as possible.) Fix such a series for S. Suppose S has qftl conponents, q _>_ 1. S/Sp has property (3) so it has a principal series S/Sp'Sé/SPD Si/SPD‘“DSI')/Sp- Sp/Sp’ with compon- ents (Ik/Sp)/(Ik+l/Sp), k = O,l,...,(q—l), each having property (3) by the induction hypothesis applied to S/Sp. Then (Ik/Sp)/(Ik+1/Sp) g Ik/Ik+l’ and a principal series for S can be chosen with the Ik/Ik+l’ k = O,l,...,(q—l), as its components, namely 8 = SéDS'D-HDS' - S :S +13"'Dsr+1‘ This series still satisfies 1 p p p the maximality condition on p. Thus, all that needs to be done is to prove that the last component, Sp/Sr'tlg Sp’ has property (3). If p - r, this is true, for then sp- Sr is a group (or perhaps a group +1 is a group, say with identity e, so 5 = eseSs for each seSp-S with zero s'nceS S i nn-ull. Thu assume < . -S )1 p/p son 5, prSpp'u p+l. If Sp does not satisfy property (3), let k be the minimum positive integer, k > p, such that there is an as SIC-skid With a¢85p- Sk/Sk4rl is a null factor by the chalce of p. Consider SpDn-DSkD-uDsfll. Sk-Skd -= Ja is the J-class containing a, by a remark in the prelim- inaries. That Sp-Jas Sp-(Sk-Sk "1) is an ideal of S will be proved. Then a new principal series can be formed for S which contradicts the 23 maximality of p. In fact, Sm'Ja = 3111- (Sk—S ), p Sm < k, is an k+1 ideal of S. For let be Sm-(Sk— S ) and suppose there is an s in 8 kid such that bse Ja' (Sk- sk+1)' If be Sk+l then bse 8k“; sm’(sk- Ski-l) as desired, so assume be Sm- Sk' be 5111- 5k and m < k imply bebSp by the minimality of k. Let ce Sp such that b =- be. bse J a' 51,- Sktl means bs and a are in the same J—class. But 5 is commutative and has property (3) so be and a in the same J—class means (bs)S = aS. Since as as there is an s' in S such that (bs)s' =- a. But b a be and bss' - a imply a - (bs)s' ' (bc)ss‘ = (bss')c = ac with ceSp, which contradicts the assumption that a é asp. Then she Sm- Ja 8 Sm-(Sk-Sk*l) for each s in s and sm-(sk-sk,1) is an ideal of s for p Ssm <1k. Let Rm- Sin-(sk-Sktl)’ p S m < k. Form the ideal series 5-6513 spv- 055-1 Sig-RP: Sp'(sp'(Sk-sk+l)) = Sk’sk4-1 is a J-class of S since Sk/sk+l is a principal factor of 3. Also SIB/RP“ sk/s1M1 is null. ' B on. DSp SpDRpD DRk—135r+l for S. Then Rm’le = (Sm-(Sk-Sk4-1))-(Sm+l'(sk-Sk¢l)) *3 Sm.sm+l is also a J-class of S for m = p,...,(k-l), so all factors of this series except perhaps R'k-l/Sr *1 are principal factors (see the preliminaries). Note that R = S -S is non-null and is the (p+l)st p-Rp+1 p p+l so Rp/‘Rpit-l factor of the series. Then the maximality of p is contradicted by N Sp/Sp.1 refining this series to a principal series for S. Thus Sp must have = ' ' .0. ' 00. . had property (3). Then 8 803513 DSp‘SPDSpflD Dsr+l is a principal series for S with components which have property (3). Corollary 1: A commutative semigroup with property (3) has a principal series for which each component is a semigroup of type C. 24 Semigroups oftype C' are examples of what Ljapin [6, p. 249] calls semigroups with s_eparating group art. Such a semigroup is one which has an identity and in which the product of We elements which are not two-sidedly invertible is not two-sidedly invertible. (An element is two-sidedly invertible in S if it is a right and a left divisor of each element of S [6, p. 218].) That a semigroup of type C' has an identity can be proved as follows: lemma 3.6: Let S be a semigroup of type C' with corresponding group G. Then the identity of G is an identity for S. m: This is clear if S is a group. The proof is by induction on r, where S = $035 D-HZZDSr,1 is a principal series for S with 1 S-Sl=G. The result is clear if r = 0. If r = l, S =- $0381352= (O) and Sigsz ' (G) as $1 is nilpotent. Then since S has property (3), if as 51-52 there is a geSo-Sl such that ga = a. If e is the identity of G, ea - e(ga) - (eg)a = ga = a. In like manner ae = a. Assume r > 1, and suppose the lemma is true for all semigroups of type C' with principal series of length less than r+l. Homomorphic images of semigroups of type C' are of type C' by lemma 3.4 so, by the induction hypothesis, e is the identity for s/sr = (s-srm z with the identi- fication described in the preliminaries. I.e., if as 5-81., then as = se = 3. Let ae Sr’ 3 f 0. Then property (3) in S implies the existence of se S such that sa = a. 35 5-81,, for otherwise “ sa = a. In like a - sae Sr+l 8 (0). Then ea = e(sa) = (es)a — manner as = a. Thus e is the identity of S. 25 Theorem 3.1 and lemma 3.6 can be used to prove the reduction theorem stated at the beginning of the section. Theorem 3.2: Let S be a commutative semigroup. If F(S) is quasi— Frobenius then S has a principal series with components Ik/Ik+l’ k-O,l,...,q, such that F(Ik/Ik+l) is quasi-Frobenius for each k'0,1,...,q. Conversely, if S has such a principal series and I0 = S, then F(S) is quasi-Frobenius. .EEEEE: Suppose S has a principal series with components Ik/Iktl’ k=o,1,...,q, and with IO=S. Suppose also that F(Ik/Ikd) is quasi- Frobenius for each k. Then F(S) is quasi-Frobenius by induction on q. If q=0, then S=Io'.'DIl and Io/If-Io'S is the only component so F(S)=F(Io) is a quasi-Frobenius algebra. Suppose the result is true for all such semigroups with less than q+l components. Assume q>O. Then F(S/Iq)¢ F(S)/F(Iq) is quasi-Frobenius since S/Iq=Io/IqD---:>Iq_1/Iq3Iq/Iq is an ideal series for s/Iq and (lk/Iq)/(Ilm/Tq)9--’Il(/:fk ,1, k=0,l,...,(q—l), are components of S/Iq corresponding to this series. F(Ik/Ik+l) is quasi- Frobenius, so by the induction hypotheses F(S/Iq) is quasi—Frobenius. F(Iq) is also quasi-Frobenius. Then lemma 2.1 together with F(S/Iq)g F(S)/F(Iq) and F(Iq) being quasi—Frobenius imply F(S) is quasiéFrobenius. Conversely, suppose F(S) is quasi—Frobenius. Then S has property (3) by lemma 3.1 so it has a principal series with components Ik/ikwl’ k=0,1,...,q, of type C by theorem 3.1. Also IO=S by lemma.3.2. Lemma 3.6 implies the existence of an identity for each component. The proof that F(Ik/Ik+l) is quasi-Frobenius is also by induction on q. If qeO; then S = I023 Il implies F(S) g F(Io/Il) is quasi-Frobenius, as Il - (GO or I1 is empty. Thus, assume q 2:1 and suppose the result 26 is true for all non—negative integers less than q. Let f be the identity of F(Iq) t! F(Iq/Iqfl), and let e be the identity of F(S). Then F(S) g F(S)(e—f)¢F(S)f=F(S)(e-f)¢F(Iq) as an algebra direct sum, since F(S)f=F(Iq) (because F(Iq) is an ideal of F(S) with identity f.) Then F(Iq) and F(S)(e-f) are direct summands of the quasi-Frobenius algebra F(S) so they are quasi-Frobenius as observed in the prelim- inaries. But F(S/Iq) g’F(S)/F(Iq) “ F(S)(e-f) quasi-Frobenius implies, by the induction hypothesis applied to S/Iq, that the components (Ik/lq)/(Ik*1/lq) of S/Iq have quasi-Frobenius algebras over F, k=O,l,...,(q-l). Thus F(Ik/Ikfl) is quasi-Frobenius for k=O,l,...,q. Precisely the same proof can be used for Frobenius algebras. Theorem 3.2 is the central theorem of this development. All the results of sections 4 and 5 are devoted to finding conditions under which a semigroup of type C has a quasi-Frobenius algebra. These results should be viewed with this theorem in mind. 4. Semigroups of types C, C', and D and their algebras. Theorem 3.2 reduces the problem of characterizing commutative semigroups with quasi-Frobenius algebras to that of describing semigroups of type C which have such algebras. Let S be a semigroup of type C with J-classes J i = O,l,...,(r+l). In this section it i’ is found that the set (J1: i = O,l,...,(r+l)) forms a semigroup T of type D, relative to the products of the complexes Ji of S. Call T the Jeclass semigroup corresponding pp'§ and let G be the group corresponding to S. The main theorems of this section are: Theorem: If S is of type C and F(S) is quasi-Frobenius, then F(S) is Frobenius. Theorem: Let S be of type C with group G and J-class semigroup T. If (c(F),o(G)) = l and F(S) is quasi-Frobenius, then F(T) is quasi- Frobenius. This theorem indicates that semigroups of type D should be studied. The following result is useful. Theorem: Let T be a semigroup of type D with a principal series JFDo-oibT *1 = (0). Then F(T) is quasi-Frobenius if and only r if r(F(T1)) = F(Tr). ' The goals of the section then, are first, to reduce the ”type C T = T EDT 0 problem" to the investigation of semigroups of type D, and second, to find semigroups of type D which have quasi-Frobenius algebras. In section 5 the process is reversed. There a method of constructing 27 28 some semigroups of type C from arbitrary abelian groups and arbitrary semigroups of type D is given. If S is a semigroup of type C' with group G, then F(G) can be considered a subalgebra of F(S). This will be done. In what follows S is a semigroup of type C' (at least) with a principal series 5 = SODSID-HDS Sr+1= (0)) if S has a zero and Srtl is empty r+l’ otherwise, 5-813 G, the group of S, and 31 is nilpotent. Let Ji’ 1 8 O,l,...,(r*l), be the (1-0135338 or S, J 8 Si-si‘J,’ i : O,l,...,I‘, i Jr+l' (O) or is empty. The structure of S is greatly influenced by G. Lemma 4.1: Let S be a semigroup of type C. If a,be Ji’ OSiSr+l, and a '- sb, then s is in G, or a = b = 0. Then SfLSngj'.’1 if r+l > 3 2 i 2 1. Proof: Let a,be Ji' Then Sa = Sb since S is of type C, so there are s,s'e S such that a *- sb and 3's = b. Then a = sb = (ss')a so a = (ss')ka for each positive integer k. If either 8 or s' is in 51’ then ss'e S1 and $1 is nilpotent so a = b = 0. Thus either s,s'€ G ora=b=0. 2 . . SiS C_:_S is clear if i = j __>_ 1, for then Signs,”1 Since $1 is 3 3+1 nilpotent. If r 23 > i 2: 1, let aeSi, be Sj' Then abeSj smnce Sj . d. is an ideal of S. If b i! Sj.sj+l’ then b653,.l and abesj‘tl as deSlre .. ° - f Thus suppose beSj-Sjfl - JJ. Then as Jj is a J class and S is 0 type C, Sab = Sb. This implies the existence of an as S such that s(ab) = b. Then (sa)b = b so (sa)kb = b for each positive integer k. But 31 is nilpotent so if b 7! o, then as: 5-51: G. But as Ji’Si’si +1, 1 > 0, so a contradiction arises. Thus abeSj*1. 29 The following observations will be needed. Lemma 4.2: let S be a semigroup of type C. Let geG. Then 1) g5 " 5. ii) ga 81gb if and only if a = b, iii) gJi= Ji’ for each i = O,l,...,r+l, iv) if a,beJi, then ga = a implies gb = b, v) G is transitive as a permutation group on the set Ji’ 1 = O,l,..,r+l, vi) if G1“ (860: ga = a, aEJi), then o(G) = (o(Gi))(o(Ji)), i = O,l,...,rtl. M: i) will be a consequence of iii). ii) ga = gb implies g-1(ga) = g-1(gb) so a = b, since the identity of G is the identity of S by lemma 3.6. then as g-lSi¢lC_:S a contradiction. iii) If aEJi and gas S. 1+1, 1+l’ Thus gae Ji' Then gJigJi. Then, by ii), gJ,L = Ji since o(Ji) is finite. iv) Ji is a J-class of 8. Thus if a,beJi then sa = b for some 363 since Sa = Sb and beSb by property (3). Then gb = g(sa) = s(ga) = sa 8 b. v) G is a permutation group on Ji by ii) and iii). Let a,b€ Ji' Then as . bS since Ji is a J-class of S, so there is an 568 such that sa = b. But sa = b implies as G by lemma 4.1. Thus G is transitive on the set Ji' vi) Since G is transitive on Ji and an element of G either fixes each ae J. or moves every such a, (o(Gi))(°(Ji)) ’ o(G) by [10, Theorem 3-2, 1 p. 5]. 30 In what follows, the Gi will have the same meaning as above, and S will be a semigroup of type C unless otherwise specified. lemma 4.3: i) If aeJi and aJJan is nonempty, then an= Jk’ < ' < o . . (D _ i,J _ r 11) If aeJl, b6 J3 and she Jk’ then GingGk. Proof: 1) Let b,b'e Jj and abeJk. Then there is a ge G such that b' 8 gb by lemma 4.2 v). Then ab' = a(gb) -= g(ab)e Jk by lemma 4.2 iii). Thus aJ QJ . Let cgceJ 3 k k’ c there is an h in C such that he = c'. Then a(hb) = h(ab) 8 ho a c' = ab. By lemma 4.2 v) and hbeJ by lemma 4.2 iii), so JkgaJ . Thus an= Jk‘ J 3 ii) Let geGi, héGJ. and abe Jk’ say ab = c. Then (gh)c = (gh)(ab) - (ga)(hb) = ab -= c. Thus Giojgck. RM: As a result of lemma 4.3, the set T 3 (Ji: 1 = O,l,...,r.-l) is a semigroup relative to the multiplication of the complexes J1 of S. T is a homomorphic image of S via the following congruence R: Define aRb if and only if aS = b3, a,beS. This is just the usual equi- valence relation on S which yields the J-classes. But in the com- utative case it is also a congruence. For as = b5 certainly implies c(aS) = c(bS) or (ca)S = (cb)S and (ac)S = (bc)S for each c in S. This homomorphic image T of S was~ approached in an'around-about way because the lemmas concerning the products of the J—classes Ji will be used later. T is closely related to S in some ways. lemma 4.4: There is a one-to-one correspondence between the principal series for S and those for T. T is of type D. Proof: Let S-'S/R - T be the natural homomorphism with s—asR. If S . 503513...35r+l is a principal series for S with J; iii-51,1, 31 then S/R = So/RDSI/RD-"DSfil/R is a principal series for S/R = T, and conversely. S/R = T is of type C as the homomorphic image of a semigroup of type C is of type C.by lemma 5.4. But si/h —- Sin/R = (Si-Shiva “ Ji/R are the J-classes of S/R e T, and Ji/R consists of precisely one element since a,be J i.if and only of aRb. Before continuing with the connection between S and T and their algebras, the relationship between F(G) and F(S) is examined. As O is a subset of S, F(G) can be considered a subalgebra of F(S). Then F(S) =- F(G) + F(S), as algebras, (this is not a direct sum since F(G) is not an ideal of F(S) in general). F(Sl) is a nilpotent ideal of F(S) so F(Sl) is in the radical of F(S). Let N be the radical of F(G). If (c(F),o(g)) = 1, then N = (0) since F(G) is semisimple [5, Theorem 16.4.1, p. 257] so the radical of F(S) is F(Sl), and F(S/sl) ’=‘-’ F(S)/F(sl) = F(G). Lemma 4.5: The radical of F(S) equals F(Sl) + (radical (F(G))). £11993: Let M denote the radical of F(S), N the radical of F(G). F(Sl) is an ideal of F(S) and F(Sl);ll. Let h: r(s)—~F(s)/F(sl) be the natural homomorphism. Then (l)h is the radical of (F(S) )h = F(S)/F(Sl). But (F(S))h = (F(G) + F(Sl))h = (F(G))h, so (M)h -- (N)h and thus 11 = N + F(Sl)' Idempotents play an important role in the decomposition of algebras so the following lemmas are of use. lemma 4.6: Let S be a semigroup of type C. If f is an idempotent of F(S) then f€F(G). 32 Proof: Suppose f = a+b, a€F(G), be F(Sl). Then f2 = a2+2ab+b2= ab and a2- a = b(e—2a—b). But (a2 - a)e F(G) and b(e—2a—b)€ F(Sl) so a2- a and b(e—2a-b) =- 0 by the linear independence of the semigroup elements over F. Then (e-2a)b 8 b2. b€F(Sl) so b is a linear combination of elements from 81. If b 7‘ O, let i be the minimum positive integer such that the expression for b contains an element ' of Ji with a nonzero coefficient. Let b *3 b1+ b2, b1(7‘ O)e F(Ji), b2 5 F(S Then (e-2a)b1= 0 since (e—2a)ble F(Ji) and (e-2a)b2E F(Si+l)' i+1)’ If the characteristic of F is 2 then b1: 2abl= O and a contradiction arises unless b = 0 as desired. If the characteristic of F is' not 2, 1=~2a(2abl) = 4a2bl- 4ab1, so bl: 2ab1= 0, a contradiction. Thus b =- O and f = aeF(G). then 2ab Then, in particular, the primitive idempotents of F(G) are primitive in F(S). This can also lbe proved for C'-semigroups. Ie:mma 4.7: Let S be a C'-semigroup. Then each primitive idempotent in F(G) is primitive in F(S). Proof: Let f be a primitive idempotent in F(G). Suppose 2 . . . g f f1+ f2, flf2= f2f1=O, fi fi, i 1,2, 111 F(S). Let fi ai. bi, . 2_ . . aie F(G), bie F(Sl)’ l 1,2. Then fi- fi implies 2 2 . - 2: ais- aib 1+ biai+ bi: ai+ bi‘ Then by linear independence, ai ai, i - 1,2, since all other terms are in F(Sl). But a1a231=0 since f 13f2 a1a2+ a. 12b + blaZl'bl =0 and ala 2 is the only term in F(G). But f -= (af- a2) 1: (blt b2):F(G), so blt‘bz = 0 and f = al+ a2, which contradicts f being primitive in F(G). Thus f is also primitive in F(S). Thus, a decomposition of e, the identity of F(G)C_Z_F(S), into 35 pairwise orthogonal, primitive idempotents of F(G) gives rise to a decomposition of F(S) into a sum of principal indecomposable modules when S is of type C. Then every principal indecomposable module of F(S) is isomorphic to one generated by a primitive idempotent in F(G). Before continuing with this, some remarks should-be made about the decomposition of an arbitrary finite dimensional associative algebra A with an identity e [8, p. 612]. e can be written as a sum of pairwise orthogonal primitive idempotents eij’ and suppose these idempotents are enumerated as follows: 9 8 el’1¢ 81’2+000* el’f(l)* 82,10...* 82,f(2)+...¢ en,l+00°* en,f(n)’ g ' a ' where Aek’l new for 1 l,...,f(k), and Aek’ii' hand if k f m. Let eke ek,1, k 8 l,...,n, be “representatives” for the n pairwise non-isomorphic principal indecomposable modules (left)‘Aek i? Lek, ’ i = l,...,f(k); f(k) is the number of Apisomorphic copies of Aek which appear in the decomposition, A‘Ael, 1‘ o o o‘Ael,f (1)+132,l*0 o .*Aea,f (2 )* o . 0+Aen,1* o o o+ken’f (n) . (Note: These remarks can be made for right Apmodules also.) Let a-—+a' be the homomorphism of A-%A[Radical A. Then one has the following characterization of quasi-Frobenius and Frobenius algebras A. [8, Lemma 2, p. 6l4] Theorem (Nakayama): Let A have an identity. A is quasiAFrobenius if and only if there is a permutation p of the set (l,...,n) such that for each k in (l,...,n): ekA has a unique irreducible right A—submodule rk and i) g 1 1 - rk ep(k)A as right Aamodules, 54 ii) Ac (1:) has a unique irreduCible left A-submodule lp(k) and l 113(k)“ A' ek as left A—modules. Furthermore, A is Frobenius if and only if there is a permutation p Which also satisfies iii) f(k) - f(p(k)) for each k in (l,...,n). Returning now to semigroup algebras, let S be a semigroup of type C with group G. Let F(G) play the role of A in the discussion above, and let rk be the unique irreducible right F(G)-submodule of eka) with rk" e'(k)F(G)'. F(G) is Frobenius so such modules and such a permutation exist and satisfy i), ii), and iii). Recall that the radical of F(S) is F(Sl) + (radical F(G)). The next lemna indicates where the irreducible F(S)—submodules of ekF(S) can be found. Lenma 4.8: Let ek be a primitive idempotent of F(G);F(S). If se S such that e ksrkf (0) and eksrksl= (0), then eksrk is an irredu- cible right F(S)-subuodule of e kF(S) and e kksr 5e ep'(k)F(S) as right F(S)—modules. Proof: (eksrk)F(S) = e kksr (F(G) 4- F(Sl))= eksrkF(G) = eksrk since eksrksjf (O) and r kF(G)--- rk. Thus, eksrk is a right F(S)-sub- module of ekF(5). e sr is irreducible. For let xe eksrk, say 1 = ekSy, yerk. k R Then xF(S) = eksy(F(G) 4- F(Sl)) = eksyF(G) *3 eksrk, as ye rk and rk is an irreducible F(G)-module so yF (G) - rk. g 1 v ‘8 t ‘ Next, eksrk ep(k)F(S) . Note first that e' 1300 F‘(S) e POOFQG) as right F(S)-modules if one defines for ue F(G) and x = 11* x2 in 35 F(S) I F(G) + F(Sl), x16 F(G), x26 F(Sl), (ep(k)u)‘x = (ep(k)ux)' = (ep(k)uxl)', since F(Sl)CRadical of F(S) so (eI')(k)u')x2 - (rad.F(S))' = 0'. Let u: rk—o(ep(k)F(G))' be the F(G)—isomorphism insured by F(G) being quasi—Frobenius. u will be used to construct an F(S)-iso- morphism of eksrk onto (ep(k)F(G))' - (ep(k)F(S))'. Define O: eksrk—fi(ep(k)F(G))', as right F(S)-modules, with (eksu)c = uu, ue rk. a is well-defined and one-to-one as: If eksu = eksul, u,u1e;rk, then eks(u-ul) - 0. Then 0 = (eks(u-u1))a ' (u—u1)u - uh - “1" so u = u1 since fl is an F(G)—isomorphism. c'is an F(S)-homomorphism. It is clearly F-linear. If x = xlf x2 is in F(S) - F(G) + F(Sl)’ then ((eksu)x)a = ((eksu)x1)a (since eksrkSl= (O) and uerk) (ux1)n - (un)x1 (since v.13 an F(G)-homomorphism) (un)(x1¢ x2) ( by the definition of (ep(k)F(G))‘ as a right F(S)-module) = (un)x = ((eksu)a)x (by the definition of 0). Thus a is an F(S)-isomorphism. The same process can be carried out for the imique irreducible F(G)-submodule lp(k) of F(G)ep(k). It is reasonable to ask whether F(S) quasi-Frobenius implies F(S) is Frobenius. The next lemma will be used to prove that this is the case for semigroups of type 0. Since the decomposition of e into mutually orthogonal, primitive idempotents of F(G);F(S) yields a corresponding decomposition for F(S) into principal indecomposable 36 modules, the condition f(k) = f(p(k)) in F(G) forces one to ask if it holds in F(S). I.e., when are two principal indecomposable modules e1F(S) and 92F(3) isomorphic; Bi, 1 = 1,2, primitive idempotents of F(G)? Lemma 4.9: If 31 and e2 are primitive idempotents of F(G)g;F(S), then elF(G) 3 ezF(G) as right F(G)-modules if and only if elF(S) “ e2F(S) as right F(S)-modules. Erggf: In general, two principal indecomposable modules 61A and 32A are isomorphic if and only if el/el(rad.A) g ez/e2(rad.A) [3, Theorem 54.11, p. 572]. Also eiA/ei(rad.A) 9* (eiA)' as right Armodules, under the isomorphism eia + ei(rad.A)-»eia ¢ (rad.L), a6 A, i - 1,2. In particular, eiF(S)/ei(rad.F(S)) 9' (eiF(S))' = (eiF(G))', where (eiF(G))' is considered a right F(S)émodule as in the proof of lemma 4.8. Thus elF(S) ‘:‘=' e2F(S) if and only if (e11«‘(o))'g (e2F(G))' as right F(S)-modules by the remarks above. But if the Operators are restricted from F(S) to F(G), then (elF(G))' z (e2F(G))' as F(S)-modules if and only if they are isomorphic as F(G)-modules. Then using the result stated above, (elF(G))' g (e2F(G))' as right F(G)-moduhes if and only if e1F(G) and e2F(G) are F(G)-isomorphic. Thus e1F(S) z e2F(S) as F(S)-modules if and only if elF(G) g e2F(G) as F(G)-modules. The same result holds for left modules since F(S) is commutative. Theorem 4.1: If S is of type C and F(S) is quasi-Frobenius, then F(S) is Frobenius. 57 BZEEE‘ Suppose F(S) is quasi-Frobenius. Let e = el,l+...¢ el,f(1)* e2’1+...+ e2,f(2)+...+ en,l*"’+ en,f(n)’ be a decomposition of the identity e of F(S) into a sum of pairwise orthogonal primitive idempotents ek,i of F(G)C;F(S) (this can be done as ee GCF(G)<;F(s)), with ek,1F(G) 3' ek,iF(G) for i =- l,...,f(l), k - l,...,n, and ek’iF(G) 79m,3F(G) if k 71 m, as before. The ek’i are primitive in F(S) by lemma 4.7. ek,iF(S) 9 em,jF(S) if and only if ek,iF(G) 9’ em’jmo) by lemma 4.9, so ek,iF(S) a em,jF(S) if and only if k = m. Let ek= ek,l’ k - l,...,n. As F(G) is Frobenius, the permutation p of Nakayama's theorem can be chosen so that f(k) 8 f(p(k)) in F(G). But this same permutation can be used for F(S) by the preceding remarks, so f(k) = f(p(k)) in F(S). Since F(S) is quasi- Frobenius, conditions i) and ii) of Nakayama's theorem are also sat- isfied, and so by that theorem F(S) is Frobenius. Continuing now with semigroups of type C, let ek be a primitive idempotent of F(G);F(S). If seS, eks f O, and ekssl= (0), then eksrk is an irreducible right F(S)-submodule of ekF(S) if rk is the unique irreducible F(G)-submodule of ekF(G), and eksrkg (ep(k)F(S))' as right F(S)—modules, by lemma 4.8. If (c(F),o(G)) = 1, then r = ekF(G) as then F(G) is semisimple. k Let F be fixed and let S be of type C. Define R1; (seS: eke = O). Lenma 4.10: Bk is an ideal of S for each idempotent ek of F(G). Let s eJi, lot g1,...,gm, be coset representatives of G1 in G, and let es 2ag,aeF. ThenseRkifandonlyif 2 a =0foreach k ge G g g g e ngi j = 1,...,m. 58 Proof: B‘k is an ideal of S. For if 56 Rk and 5's S then e s = O k implies ek(s's) = ek(ss') *3 (ek s)s' = 0. Let seJi. sek==§n:l 2 asg= 2( 2 a)sg. since J=lg€gGg 3=1g€gGg J J J l -l sg = sg' if and only if g g'e Gi’ by lemma 4.2. Then sek= 0 if and only if 2 a = 0 for each 3' = l,...,m, by the linear independence as go G j i over F. These ideals Rk can be used to get a sufficient condition for a semigroup of type C to have a quasi—Frobenius algebra. Before proving this theorem note the following. The ideal series S=SOD SIQRRQW) can be refined to a principal series for S, say S'soDSlD-“DS“1=(O), with 5M a Rk. Then if ses-sp l=S-B.k,e ks 7‘ o. If as sp-spd, then eka 7‘ O and ekaSl= (0) since laS‘gng, as in lemma 4.8. Keeping this notation one has: Leanna 4.11: If F(S) is quasi-Frobenius, (c(F),o(G)) = l, and .- 3 — h eS'” 0 aeSp S Sp Rk’ then aesS for eac s R‘k p+l Proof: If se (S—Rk), then eks 7! 0 so eksF(S) 7! (0). Then eksF(S)QekaF(G) because ekaF(G) is the unique irreducible right F(S)-submodule of ekF(S) by lemma 4.8, and ekF(G) -- rk is the unique irreducible F(G)—submodule of e kF(G) since (c(F),o(G)) = 1. But then e aeek sF(S) and e, the identity of G, appears as a summand in ek so k a appears as a sunmand in eka. Then there must be an element s'e S such that a = 33', by the linear independence of the nonzero elements of 8 over F. This is a necessary condition for F(S) to be quasi-Frobenius when (c(F),o(G)) = l. A sufficient condition follows. 39 Theorem 4.2: If F is of type C and if F(S/Rk) is quasi-Frobenius for each k in (l,...,n), then F(S) is quasi-Frobenius. Proof: One need only show that ekF(S) has a un'gue irreducible right F(S)-submodule for each k in (l,...,n) and then lemma 4.8 implies the rest. Suppose ekxF(S) is an irreducible F(S)-submodule of ekF(S). Then ekxsl' (0) so ekxF(S) = ekxF(G). Then ekxF(G)§;F(SéRk) since ekRk. (0). Let h: S-—+S/Rk be the natural homomorphism of these semigroups and extend it linearly to a homomorphism h: F(S)-+F(S/Rk), i.e., ( 2 a s)h - 2 a (sh). Then (ekxF(G))h is an irreducible 368' s as S' S F(S/Rk)-submodule of (ekh)F(S/Rk). Thus, if ekxF(G) and ekyF(G) were two irreducible right F(S)-submodules of ekF(S), then i) ekxF(G) and ekyF(G) are contained in F(S—Rk), and ii) (ekxF(G))h = (ekyF(G))h. But ii) implies ekxF(G)gekyF(G) + F(Rk) which is impossible by i) unless ekxF(G) = ekyF(G). Thus ekxF(G) is unique for each k. Then by lemma 4.8 and Nakayama's theorem F(S) is quasiéFrobenius. More information about semigroups S of type C is needed. Some insight into the structure of S and its algebra F(S) is provided by studying the relationship between S and T, its J—class semigroup. The next theorem will show that if (c(F),o(G)) = 1, then F(T) is quasi- Frobenius if F(S) is. The notation is the same as before; i.e., S is a semigroup of type C with group G and Gi’ i = O,l,...,rel, are the T is the "fixing groups” for the J-classes Ji’ i = O,l,...,r+l, of 5. corresponding J—class semigroup. Theorem 4.3: If (c(F),o(G)) = 1, then F(T) g F(S)E, where E = (0(0) $12 g, F(T) is isomorphic to a direct summand of F(S), and giEG F(T) is quasi-Frobenius if F(S) is quasi—Frobenius. Proof: Let (G:Gi) = o(Ji) = ni, (G:Gi) the index of G1 in G, and let o(Gi) 8 mi, i = 0,1,...,r+l. Then.mini= o(G) = no. (This is justified by lemma 4.2 vi).) Let E; =- 2 s, i = O,l,...,r+l. se.L i m E... 2 - ’12 - - en 1 5 mi gsi, 51 a fixed element of J., by lemma 4.2 1v). seJi geG l The proof will consist of showing that the elements Ei= uglsi , i = O,l,...,r+l, form a semigroup which is isomorphic with T, are con- tained in F(S)E = F(S)Eo, and are linearly independent over F. -1 v 3 Consider SjEi mi :2 gs geG If s.sieJk, then .s.. J l J (l) 83E; = mil :Eégsk = mkmElEfi since gsk= hsk if and only if ge h’lg e ck. Then (2) EBEi - 2 sjEi = 2 mkalEl'c == njmkm'i'lfiil'< by using (1) and 83: Jj SJ er the distributive law. Then. EjEi= (njni)-1E3Ei , by the definition of Ej and Ei’ = mk(nimi)-lEl'(’ by (2), = - ' since n.m.= n mknolEk, l l 0’ . -l_ = nilEi = Ek’ by the definition of Ek and Since mkno - nk . Thus, the set (Ei: i = 051,...,r+l) is a semigroup. The elements E., i = O,l,...,r, are linearly independent over F since they are sums i of elements from distinct J-classes of S and the nonzero elements of S are linearly independent over F. That Ei-»Ji, i = 051,...,r+l, is an isomorphism of the semigroup onto T is clear since, again, Ei is the sum 41 of elements from Ji so the El multiply as the Ji do. By (1), 31E; ' mimSlEi - miEi since mo- 1, so siEoB E1. Thus Bic F(S)Eo- F(S)E for each i = O,l,...,r+l. These conditions imply F(S)E g F(T). That F(S) g F(S)E + F(S)(e -E) is a direct sum follows from E(e - E) = 0. Then F(T) g F(S)E is quasi-Frobenius if F(S) is, since F(S)E is a direct sunmand of the quasi-Frobenius algebra F(S). I The converse of theorem 4.5 is false. Example (1) in the appendix is a counterexample. Thus semigroups T which are of type D need to be studied. The remaining part of this section will be devoted to such semigroups. That each semigroup T of type D does occur as a J—class semigroup for some semigroup of type C is obvious since T itself is of type C. Later a method will be given by which other semigroups of type C can be con- structed from T. Let T be an arbitrary semigroup of type D. Let T = TOZDTlZM-CDTr*l be a principal series for T, with Ti- Ti+l= (ti), i = O,l,...,r, Tr¢l. (0) if T has a zero and Tr+l empty otherwise. to is the identity of T and (to) = T‘- T1 is the group corresponding to T. T1 is nilpotent. If F is a field, F(Tl) is the radical of F(T) by lemma 4.5. tO is the only nonzero idempotent of F(T) by lemma 4.6, so tOF(T) = F(T) is the only principal indecomposable F(T)—module. Then F(T)/F(Tl) a F(to) % F is the only irreducible F(T)-module. Lemma 4.12: If F(T) is quasi-Frobenius then tre tT for each t(%o) in T. (I.e., each nonzero element of T divides tr.) If t6 T such that Tlt = (0), then to Tr' In addition, if T is the J-class semigroup for a semigroup S of type C then S has a J—class Jr, each element of which is divisible by every nonzero element of S. M: If F(T) is quasi-Frobenius, Nakayama's theorem implies F(T) has a unique minimal ideal or irreducible submodule, since F(T) is itself the only principal indecomposable module. F(Tr) = F(tr) is a minimal ideal of F(T) since Ttr= Tr= (two). If t(;lo) is in T, then F(tT) is an ideal of F(T) so it contains a minimal ideal of F(T). Thus F(tT):_DF(tr) so tre tT for each t(740) in T. If te T and Tlt = (0), then Tt = (t,0) and F(t) is a minimal ideal of F(T). Then tGTr. Let S be a semigroup of type C and let T be its corresponding J-class semigroup, G its group. If (c(F),o(G)) =3 l and F(T) is quasi-Frobenius, let Ji be the J-class of 5 corresponding to ti in the homomorphism of 5 onto T, i = (D,l,...,r+l. Let aeJr, s€ Ji’ i S r. In T there exists a tk such that tkti= tr by the first part of the lemma. Then JkJi= Jr' But lemma 4.3 i) implies Jks = Jr so there is an 5' € J such that s's = a. This can be done for every such as Jr k and for each 36.11, i = O,l,...,r, so each element of Jr is divisible by every element (nonzero) of S. Lemma 4.12 does not provide a sufficient condition for F(T) to be quasi—Frobenius. Example (2) of the appendix is a counterexample. It is, in some sense, related to a sufficient condition however, in that F(T) is quasi—Frobenius if and only if r(F(Tl)) = F(Tr)' Before proving this, consider the following observation which is used several times in proofs . 45 Lemma 4.13: If x€F(T) and te Tl, t 7’ 0, then xt -- o» implies xe F(Tl). Proof: If x = etc-t x', as F, x'e F(Tl)’ then xti= ati+ x'ti= 0 implies a = 0 by the linear independence of the nonzero elements of T over F. For by lemma 4.1, since x'e F(Tl)’ x't€F(Ti*l) but ati is Theorem 4.4: F(T) is quasi-Frobenius if and only if r(F(Tl)) = F(Tr). .Prggf: F(T) is quasi-Frobenius if and only if F(Tr) is the unique minimal ideal of F(T) by lemma 4.8 and Nakayama's theorem since F(T) is the only principal indecomposable module for F(T) and F(Tr) is a minimal ideal of F(T). If xeF(T), then xF(T) is a minimal ideal in F(T) if and only if xe r(F(Tl)). For xF(T1)$xF(T) by lemma 4.1, so xF(Tl) - (0) if xF(T) is minimal. Conversely, if xF(Tl) ‘ (0), then xF(T) = xF(to) + xF(Tl) = xF(tO) = F(x) is a minimal ideal in F(T). Then F(T) is quasi-Frobenius if and only if xF(T) = F(Tr) for each :xe r(F(Tl)). Thus, F(T) is quasi-Frobenius if and only if F(x) = F(Tr)’ i.e., xe F(Tr) for each xe r(F(T1)). It is of interest to point out the broader context into which theorem 4.4 fits. In general, if A is an algebra and N is its radical, then r(N) is called the left socle of A and is the sum of all the irreducible left ideals of A [3, p. 594]. If f is an idempotent in A, then the dual module [5, p. 594] for the left A-module Af/Nf is iso- morphic to the right A—module fr(N) [3, Lemma 58.4, p. 595]. A is quasi-Frobenius if and only if, for each primitive idempotent f, the dual module of Af/Nf has composition length one; i.e., fr(N) has a composition series of length one as a right Aamodule [3, Theorem 58.6 (ii), p. 396]. If A =- F(T), then N = F(Tl) and to is the only primitive idempotent of F(T) so F(T)to/Nto9=‘ F(to) 2!- F is the only irreducible left F(T)-module for F(T), as observed before. Its dual is tor(N) = r(F(T1)). But r(F(T1));2F(Tr) so if r(F(Tl)) is to have composition length one, r(F(T1)) = F(Tr) must hold. It would be useful to have the following corollary for semigroups in general but it has not been proved nor has a counterexample been found. Corollary 1: Suppose Tlt = 0 implies teTr. If l(r(F(L))) = F(L) for each principal ideal L of T then F(T) is quasi-Frobenius. Proof: Suppose xeF(T) and xF(Tl) = (0). Then by lemma 4.13, r X€F(Tl) SO F(T)X;F(T1)O Let x z Elaiti, aie F. If X # F(Tr)’ let j be the minimum positive integer such that ajf'o. Then j <‘r. There exists a k, j < k < r, such that akf 0, for otherwise x - a.t.+~artr and (O) = Tlx'= Tl(ajtj) Since Tltr= (0), so Tltj= (0), contradicting the hypothesis. Consider F(Ttk). r(F(Ttk))§;F(Tl) by lemma 4.13, so xcl(r(F(Ttk)))2F(Ttk), a contradiction to the hypo— thesis that the annihilator condition holds for principal ideals L = 'I‘tk of T. Thus xeF(Tr) if xF(Tl) = (O) and, by theorem 4.4, F(T) is quasi-Frobenius. Conditions on a semigroup S which imply its algebra F(S) is quasi- Frobenius are difficult to find because the annihilator relations in F(S) can be very complicated even for "relatively nice” semigroups. 45 The next corollary is an example of such a condition for T. Corollary 2: Suppose Tlt = (0) implies tETr. Assume also that for each tie Tl- Tr, there exists a teT1 such that tit ,1 o and tit 7‘ tjt for each 3' 7‘ i. Then F(T) is quasi-Frobenius. r Proof: Suppose x = 2a 1it €r(F(Ti )), a,L eF. (This summation . i=1 need not start with i = 0 by lemma 4.13.) Then for te Tl’ r O=xt=2aiitt impliesaH=Owhenevertt7‘0andtijt7‘ttfor 1'1 :1 f i, by the linear independence of the nonzero elements of T over F. If i < r, there is a tETl such that tit 71 o and tit ,4 tjt for j ,1 i, by hypothesis, so ai= 0 for i = l,...,r-l, and xeF(Tr). Then F(T) is quasi-Frobenius by theorem 4.4. Theorem 4.5: F(T/Ti) is quasi-Frobenius for each i = l,...,r+1, if and only if T1 is cyclic. Proof: If T1 is cyclic, then T satisfies the hypothesis of ‘ tr+1___ corollary 2. For if Tl: (ti: i = 1,. ..,r¢l),t 0, then Tltl= (tkfi: i -= l,...,r+l) -- (0) implies k+i z r+l, for each i = l,..,r+l. k r In particular, kel Z rel so k _>_ r and tle Tr= (t1,0). Also titk= tjtk 7‘ 0 if and only if i = 3. Thus F(T) is quasi-Frobenius. l 1 Conversely, suppose F(T/Ti )- -F(T)/F(Ti ) is quasi-Frobenius for each i = l,...,r. Then by lemma 4.12, for each i = l,...,r-l, tie th for each 3’, O 5 j S i. T cyclic means each element of T1 is a power of t1, 1 say. If r = 0 or 1 then T1 is clearly cyclic. Assume r > 1 and suppose the theorem is true for all integers less than r. ConSlder T/Tr° 46 F((T/Tr)/(Ti/Tr)) F(T/Ti) is quasi-Frobenius for each i = l,...,r. Then by the induction hypothesis, Tl/Tr is cyclic. Since F(T/Trtl) ’5' F(T) is quasi-Frobenius, lemma 4.12 implies tre tT for each to T — Tr' 1 Let t6 Tl- Tr’ and suppose t' t Tl such that tr= tt'. Both t and t' are powers of the generator of Tl/Trfl (Tl—Tr)UTr so tr is also, and T1 is cyclic. In the next section, some semigroups of type C will be constructed from semigropps of type D and abelian groups. This construction will lead to other semigromls of type C with quasi-Frobenius algebras. 5. A construction for some semigroups of type C from semigroups of type D Several relationships between a semigroup S of type C and its group G and J—class semigroup T have already been indicated. T was found to be of type D in lemma 4.4. In each case G and T resulted from the given semigroup S. In this section a method is given by which semigroups of type C can be constructed with an arbitrary abel— ian group G as their group and an arbitrary semigroup of type D as their J-class semigroup. This technique will be used to extend the class of semigroups of type C which have quasi-Frobenius algebras. The idea is motivated by lemma 4.3 and the remark which follows it. There it was found that if Ji’ J3, and Jk are Jhclasses of a semigroup S of type C with corresponding fixing groups Gi’ Gj’ and Gk’ and if then GiG gGk. The procedure can be carried out in a more Jist Jk, 3 general context. Let T - (ti: 1 --O,l,...,r+l) be an arbitrary finite semigroup and let G be an arbitrary finite group. For each ti in T let Hi be a normal subgroup of G, i = O,l,...,r+l. Definition 5.1: The set (Hi: i - O,l,...,r+l) of subgroups of G is said to be admissible relative 32”: if Hinggflk whenev r titj- tk in T. In this case the notation [Hi] = (Hi: i = O,l,...,r+l) is used. It will be clear from the context what T is under consideration so the collection [Hi1'will often be called admissible. Note that any group G contains admissible collections of subgroups 47 48 relative to each semigroup T. Namely, Hi- G, i = O,l,...,rel, or Hi. (6), i -‘O,l,...,r¢l.where e is the identity of G. The main theorems of the section are: Theorem: Let S be the semigroup constructed from a semigroup T and an admissible collection [Hi] of subgroups of G. Then S is an inverse semigroup if and only if T is an inverse semigroup. Returning to commutative semigrOUps one has: Theorem: Let T be a semigroup of type D and let [Hi] be a coll- ection of subgroups of the abelian group G, which is admissible relative to T. Then there exists a semigroup S of type C with T as its J—class semigroup and with the Hi as the fixing groups for its J-classes. The following results indicate the influence that the character- istic of the field has on the algebra. They also provide examples for which the converse of theorem 4.3 is true. Theorem: Let T be a semigroup of type D and let G be an abelian group. Let S be the semigroup obtained from T and the admissible collection H03 (e), Hi. G, i - l,...,r+l, r bro. Then F(S) is quasi— Frobenius if and only if (c(F),o(G)) = l and F(T) is quasi-Frobenius. Theorem: If T is of type D, let Tl be the maximal nilpotent ideal of T. Suppose T1 is cyclic and [Hi] is an admissible collection of subgroups of the abelian group G. Let S be the semigroup obtained from T and [Hi]' If (c(F),o(G)) - l and F(T) is quasiéFrobenius, then F(S) is quasi-Frobenius. V l k The construction proceeds as follows. Let T = (ti: i = O,l,...,rel) be an arbitrary finite semigroup and let G be an arbitrary finite group. 49 Let (G,T) 8 ((g,t):ge G, téT). In the set (G,T) define: (g,t) - (g',t') if and only if g = g' and t = t',and (g,t)(g',t') 8 (gg',tt'), for (g,t) and (g',t') in (G,T). ‘With these definitions (G,T) is a semigroup. It is just the ”direct product” of the semigroups G and T. Next, let [Hi] be an admissible collection of subgroups of G, relative to Th, Define a relation R on the set (G,T) as: (g,ti)R(h,tj) if and only if i = j and g e h(Hi), i.e., geahHi. This is a congruence relation on the semigroup (G,T). It is clearly reflexive and symmetric. It is also transitive, for if (g,ti)R(g',tj) " a 3 ° 3 a ' and (g',tj)R(g ,tk), then k j i so Hk Bj Hi and g a g a g"(Hi). Thus (g,ti)R(g",tk). R is a congruence. Suppose (g,ti)R(g',ti) and (h,tj) e (G,T). a ' . 1 . . . If titj tk in T, then Hingghk so g - g (Hi) certainly implies a ' ' ' y . g — g (HR) and gh _ g h(Hk). Thus (gh,titj)R(g h,titj). In like manner (hg,tjti)R(hg',tjti). This proves that R is a congruence. Then (G,T) modulo R is a semigroup; call it 5. For convenience let [gti] = < 0. Then F(S) is quasi-Frob— enius if and only if (c(F),o(G)) = l and F(T) is quasi-Frobenius. The following lenma will also be needed in the proof. 53 Lemma 5.2: Let S be the semigroup in theorem 5.3. Let 1(I) - F(S)E. Hoof: First note that the set of elements (g—e), geG-(e) = G', is a basis for M. Forif Eaggdd, Ea = 0. Then 24g ‘3 geG g¢G g g€G g 2 a: ' ‘ ‘ » g+ae. Buta=- 2a Since Ea=050 a =' g. G' g e e g; G! g geG g g?G g8 poig'agg - (g?G‘ag) e = gEyaég—e). The set (g-e: gfiG') is clearly independent over F . l(M)2F(S)E for if E aggeld, then E 2a g - E a Eg = gCG gCG g g‘G g (Eagfil = O as 2a a O-and Eg = E for each gCG. gc G g‘G g l(M)C_-'-F(S)E. Note that if sesl, then s a (o(G))‘lss since gs = s for each s 651 (since G13 G for i > 0), so F(Sl)§F(S)E. Also F(Sl)gl(m) for if seS:L and Ea g‘ll, then 2a gs = ( Ea )s = 0 since chg g‘Gg g‘Gg gs 8 s for geG and is =0. geG g Thus suppose x - EbhhchI), xeF(G). Then x(g—e) = O for each h¢ G gee. Fist g. Then (1) o = x(g—e) = Ebhh(e—g) = Ebhh- Ebhhg. heG heG hGG Let hg =- k so kg‘l= h. Then as k takes on all values in G, h does, and 2 bhhg = 2 b _1k. Then substituting in (1), he G ke G kg 0:- Ebh— 2b k= Ebhh- 2b 1h: 2(bh—bhg_l)h hcgh keo kg’l heG h‘G hg’ h‘G conversely, so 54 for each g6 G. Then bhs b for each heG by the linear independence gh’l of the elements of C over F. Then since g was arbitrary, bh= b 1 for hg’ each gEG implies the existence of a b€F such that b=bh for each h. Thus thh - b( 2 h) = bE and 10!) = F(S)E. This completes the proof h€G heG of the lemma. .EEEEE‘ (of theorem 5.3) The condition r >'0 is imposed so that S will not be a group or a group with zero, for such semigroups have quasi-Frobenius algebras over arbitrary fields. Note that F(Sl) 9! F(Tl) since oi: G for i = l,...,r+l. Then in a principal series S = 802381 '"°:]Sr+l for S, let S-Sl ' G, and Si -Si+l= (51)“ (This is justifiable since 31- Si+l= Gti= [ti], so Si—Si+1 contains only one element. Let si- [ti].) If F(S) is quasi-Frobenius, then (c(F),o(G)) = 1 will be proved. Then theorem 4.3 implies F(T) is quasi-Frobenius. Suppose c(F) divides o(G). Let E = eEGg‘EF(G)C_'__ZF(S). Then Es = o(G)s = O for 5651‘ since 8 gs = s for each gs G and $651, and c(F) divides o(G). Eel(r(F(Sl))) will be proved. Then as E ¢ F(Sl)’ F(S) is not quasi—Frobenius. Let xer(F(Sl)), say x =- x1+x2, x16F(G), xzeF(Sl). Then ex = 32: lesx2= o with s e 51’ implies axis 0 and 3x28 0 by lemma 5.1. xlsgeEGagg, a eF, then xls '3 ( Ea )5 since gs = s for each se 51’ so xls - g (2ag=)s Oimplies Eageo. ThenEx:L sg-‘EaEg (Eagm géGg g‘G g“-‘-Gg geGg 55 Thus, x = x1+ x26r(F(Sl)) implies Ex = Exld'r Ex2= 0 so E61(r(F(Sl))) and E fr(sl). Conversely, suppose (c(F),o(G)) = 1 and F(T) is quasi-Frobenius. F(S) = F(G) + F(Sl) and F(Sl) 3' F(Tl)’ in fact T is isomorphic with a subsemigroup of S as observed before. If x = 2aggeF(G)CF(S) and 89 G r .s., thenxy = (2a )(2b 5. JJ geGg 3,13.) r y 3 2b ) since gs = s for each seSl. 3‘1 In what follows, if B is a subset of F(Sl), then let r1(B) = r(B)f]F(Sl) and 11(B) = l(B)nF(Sl). By a theorem of Nakayama stated in the prelim— inaries, the annihilaztor conditions need only be established for ideals in the radical. Thus, let L be an ideal of F(S), L.C_TF(Sl) - rad.F(S), since (c(F),o(G)) =-' 1. l(r(L)) = L will be verified. Let ll be as in r lemma 5.2. Then r(L) = M + r (L); for let x = 20.5. €F(S ) and l 131 i i l r r r r y = 2a g 4- 23.3.. Then 0 = xy = ( 2a ) Sci-sit- ( Ecisiflzajsj) gee g 3:1 3 J geG g i=1 i=1 jsl implies either x = O or 2 a; = O by the same reasoning as in the proof geG g of lemma 5.1, using lemma 4.1 and linear independence. Then also r (écisifléajsj) = 0 so Elajsj erl(x). Then Ly = (O), L§F(Sl), implies yell + rl(L). Thus r(L). = u + rl(L). Then l(r(L)) '- l(l[+rl(L)) 3 l(M)nl(rl(L)). But 1(r1.(L)) a ll+ll(rl(L)) as before. 11(rl(L)) = L since F(Sl) g F(Tl)’ and F(T) is quasi—Frob- enius (for if L' is an ideal of F(T) contained in F(Tl) then r(L')§F(Tl) 56 by lama 4-15). Thus l(r(L)) = 101) MM + L). 100 = F(S)E by lemma 5.2 so if xE€l(M)n(M 0 L), then xE = mE + yE for some 1116111 and some yeL since MCF(G) and L§F(Sl). But mE - 0. so xE 8 yEeL. I.e., l(r(L))QL so l(r(L)) - L for each ideal in the radical F(Sl) of F(S). Thus F(S) is quasi-Frobenius. Now suppose S is a semigroup obtained from a semigroup T of type D and an admissible collection of subgroups [Gi] of the abelian group G, with Go'- (e). The elements of S can be written as [gti], where [gti] - [htj] if and only if i = j and g 2 h(Gi)’ For convenience let [gti] = gti and use the same identification; i.e., gti= htj if and only if i = 3 and g a h(Gi). The multiplication is (gti)(g'tj) = htk if and only if titj= tk in T and gg' 2 h(Gk). The J—classes are, as before, Ji= Gti, i .. O,l,...,r4rl, Jo= Gtog G. Let gto= g for each geG so that G can be considered a subsemigroup of S. If T = ToDT13°"3Tr+1 is a principal series for T, then S = $02381:M-=::Sr*1'with Si-Si+l= Gti, is a principal series for S. Let F be a field. The elements of F(S) can be written in the form r 2n.t., with uie F(G)C_ZF(S), and multiplication is distributive, reducing i=0 1 1 products to those of the form (gti)(g'tj), which have already been defined. This description may seem unduly complicated but it provides a convenient way of calculating in F(S) and it will be used in the proof of theorem 5.4. T - (eti: tie T)CS, and let eti = t1 so T can be considered a subsemi— group of S. 57 Theorem 5.4: If T is of type D, let Tl be the maximal nilpotent ideal of T. Suppose T1 is cyclic and [Gi] is any admissible collection of subroups of the abelian group G. Let S be the semigroup obtained from T and [G If (c(F),o(G)) = l, and F(T) is quasi-Frobenius, then i]- F(S) is quasi-Frobenius. Proof: Let 61 be a primitive idempotent of F(G). One need only prove that eiF(S) has a unique irreducible F(S)-sutmodule for each i and then apply lemma 4.8 and Nakayama's theorem. Let Tl be generated 1 by t, tr... ' O, trf 0. Let Ri= (seS: sei= 0) as before. Suppose S-S ':»-~IDS = G, Sj-sj+l- th, is the unique principal S=SODS l l r+l’ series for S, and Sp+l= Ri (the series is unique as T1 is cyclic). Let tp€:Sp-Ri’ and let to: e, the identity of S, for convenience. If Ri= 51’ then p = 0 so eith(G) = eiF(G), and eiF(G) is the unique irreducible submodule of eiF(S) since it is unique in eiF(G) and eisl= (0). Suppose p >'l. e.th(G) is an irreducible F(S)-submodule of eiF(S) as in lemma 1 4.8. Suppose eixF(G) is another irreducible F(S)—submodule of eiF(S). eixF(G)2eith(G) will be proved. eixSl= (0.) since eixF(Sl)$eixF(G) by w r . 3 lemma 4.1, and eixF(G) is irreduCible. Let x - Eggujt , quEF(G). P . . . ~ , , J: . . Then eix = EEgeiujtJ, Since eiRi- (0) implies eit 0 for 3 >'p . q Let q be the minimum non-negative integer such that eiuqt f O. 58 p . Then 0 S q S 13 since otherwise tqéRi and eix = Eeiujtj. eiF(G) 3 J=q eiuqF(G) (since (c(F),o(G)) = 1 so F(G) is semisimple and thus eiF(G) is irreducible) so there is a v€F(G) such that ei= eiuqv. If q: = 0, then eix(vtp) = eiuqvtp - eitp, so eixF(s);2eith(c) as desired. If q >‘O, p . then multiply eix by vtp"q to get (eix)vtp—q= (2:61th3 )vtp-q' eiuqvtp .j=q - (since tJ*p-q€Ri for j > q, and then eitj*p-q' 0), which equals eitp . __ . p p . Since ei- eiuqv. Thus, again eixF(S):_2eit F(G). Then eit F(G) is the unique irreducible F(S)-submodule of eiF(S). This can be done for each primitive idempotent e i in F(G). Then, by lemma 4.8 and Nakayama's Theorem, F(S) is quasi-Frobenius. Remark: The converse of theorem 5.4 is false. Example (5) in the appendix is a counterexample. 6. Appendix Example 1. Let T be the semigroup of type D with the following multiplication table. to t1 t2 t3 t4 0 to to t1 t2 t3 t4 0 t1 t1 t2 t4 0 O 0 t2 t2 t4 0 O 0 0 t3 t5 0 0 t4 0 0 t4 t4 0 O O O O O O O O O O O F(T) is quasi-Frobenius. Let G be the group of order 2 with elements ' 2 fl . e, g, g 8 e. Let Go= Gl- G2= G3= (e), G4-G. Form S as in section 5. Then the multiplication table for S can be written as on the following page. Even if the characteristic of F is not 2, F(S) is notéqfiasi- Frobenius. The primitive idempotents of F(S) are ac: 1/2(t0¢ gto) and e1. l/2(to- gto). elF(G) is an irreducible F(G)-module. Consider elt2F(S) and elt3F(S) as in lemma 4.8. elt2F(S) I“F(tzu gt2) and elt5F(S) - F(ts- gts) are both irreducible F(S)-submodules of elF(S) since e t S - e t S = (0), by lemma 4.8, so F(S) is not quasi— l 3 l l 2 1 Frobenius even though F(T) is quasi-Frobenius. 59 60 The multiplication table for S is: to gto ti gti t2 gtz t5 3t5 t4 0 to to gto t1 gti t2 gtz t3 gts t4 0 gto gto to gt1 tl gt2 t2 gt3 t3 t4 0 t1 t1 gt1 t2 gt2 t4 t4 0 c o o gtl gtl t1 gt2 t2 t4 t4 0 o o 0 t2 t2 gt2 t4 t4 0 o o o o o gt2 gt2 t2 t4 t4 0 o o c o 0 t5 t5 gt3 0' o o 0 t4 t4 0 o gt3 gt5 t3 0 o o 0 t4 t4 0 0 t4 t4 t4 0 o o. o o o o o o o o o o o o o o o o Example 2. This is an example of a semigroup T of type D with t e,tT, for each t(# O)e T. F(T) is not quasi-Frobenius, however. r o l 2 t5 t'4 000 O O O 0 0 0 O O t t 0 0 O 0 O 0 O O O F(T) is not quasi-Frobenius since tI- tzear(F(T1)) and t1— t2 f F(Tr) - F(t4), by theorem 4.4. 61 Example 5. Theorem 5.4 says that if (c(F),o(G)) - l and T1 is cyclic, then F(S) is quasi—Frobenius, where S is obtained from T and subgroups Gi of G. ‘Without the condition (c(F),o(G)) = 1, this is false. Let G be of order 2, G = (e,g: g2= 8), let F be of character— istic 2, and let T have the multiplication table: T is clearly of type D and Tl= (t1 ,0) is cyclic. Form 5 with Go= (e), and G = G. Then S has the multiplication table: 1 t gt t o o o o l O gt? gto to t1 0 . t1 t1 t1 0 O O O O O Q This semigroup S does not have a quasi-Frobenius algebra over a field of characteristic 2, by theorem 5.3. In fact, theorem 5.5 gives other examples of this situation if T1 is cyclic. Example; 4. A reasonable conjecture might be that if T has ideals I and I such that F(T/Il ) and F(T/I2 ) are quasi-Frobenius then 1 2 F(T/Ilrilz) is quasi-Frobenius. This conjecture is false. Let T be the semigroup of example 2. Let Il= (t1,t5, t4,0) and 12* (t2,t 3,t4,0). Then I1F1I2= (t5,t4,0). F(Il) and F(Iz) are quasi-Frobenius by corollary 2 to theorem 4.4. Then T' . T/Ilr112= (to,tl,t2,z), with the 62 identification described in the preliminaries. But T1 = Tl/IlflI2 8 2 (t1,t2,0) and Ti = (z) so t1,t26 r(F(Ti)) which implies F(T') is not quasi-Frobenius, by theorem 4.4 T is also an example of a semigroup without a quasi—Frobenius algebra over F but with subsemigroups Rl= (to,tl,t5,t4,fl) and R2= (to,t2,t5,t4,0) with er1R2= (t3,t4,0) contained in T1’ such that T = Rle and F(Rl) and F(Rz) are quasi-Frobenius. Thus, decomposing a semigroup algebra into a product of quasi-Frobenius subalgebras seems to imply little about the algebra. Conjectures: 1. Let F be a finite semigroup and let F be a field of character- istic c. If F(S) is quasi-Frobenius,then every principal factor of S is either c—nonsingular or null. The converse certainly is false since every commutative semigroup is of this form. 2. Components can be defined for arbitrary semigroups just as they were defined for commutative semigroups in definition 3.2. The conjecture is that theorem 5.2 is true in this broader context. The same proof given there shows that if S has a principal series for which the components have quasi—Frobenius algebras and Io: S, then F(S) is quasi-Frobenius. The converse, however, has not been proved. . 10. BIBLIOGRAPHY A. A. Albert, Structure of Algebras, New York, 1939. A. H. Clifford and G. B.Preston, Algebraic Theory of Semigroups, Vol. I, Providence, 1961. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, New York, 1962. I J. Dieudonne, Remarks on quasi-Frobenius rings, Ill. J. Math. vol. 2 (1958) pp. 346-354. M. Hall, The Theory of Groups, New York, 1959. E. S. Ljapin, Semigroups, Providence, 1963. ‘W. D. Kuhn, Semigroup algebras, Proc. Cambridge Phil. Soc. vol. 51 (1955) pp. 1-15. T. Nakayama, 9p Frobeniusean algebras I, Ann. of Math. vol. 40 (1939) pp. 611-633. T. Nakayama, On Frobeniusean algebras II, Ann. of Math. vol. 42 (1941) pp. 1-21. H. Wielandt, Finite Permutation Groups, New York, 1964. 63