JOURNAL OF ALGEBRA 181, 371᎐394Ž. 1996 ARTICLE NO. 0125
Strong Morita Equivalence and a Generalisation of the Rees Theorem
S. Talwar*
Department of Mathematics, Uni¨ersity of York, Heslington, York YO1 5DD, United Kingdom
Communicated by T. E. Hall
Received October 24, 1994
In our earlier work we associated a natural category to a semigroup with local units and proved semigroup analogues of the celebrated Morita theorems for rings. In this article we use the notion of a Morita context to define an equivalence relation on a far wider class of semigroups. We give a semigroup analogue of Morita I and show that Morita equivalent semigroups can be constructed with ease. Our construction is based on the classical Rees theorem and a generalisation of this theorem by Hotzel. We then use properties of equivalent categories to deduce properties of this construction. Finally, we give a new generalisation of the Rees theorem and applications of this generalisation. ᮊ 1996 Academic Press, Inc.
1. INTRODUCTION
The classical Artin᎐Wedderburn theorem asserts that a ring R is simple Artinian if and only if it is the ring of linear transformations of a finite dimensional vector space over a division ring. A new insight into this theorem was provided by Basswx 2 in the early 1960s. Bass exploited a deep application of the theory developed by K. Moritawx 10 on equivalence for categories R-Mod, of modules over rings with identity. Bass’ theorem can be interpreted as asserting that a ring R with identity is simple Artinian if and only if the category R-Mod is equivalent to the category D-Mod for some division ring D. From this it was also possible to recover the Artin᎐Wedderburn theorem. The idea of such a theory is to relate a ring
* The research for this paper was carried out while the author was visiting the Laboratoire Informatique Theorique´ et Programmation, Paris, on a Royal Society Fellowship.
371
0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 372 S. TALWAR with a complex structure to one with a simpler structure and so learn more about the original ring. The machinery of Morita equivalence is in terms of projective modules, tensor products, and Hom functors. However, Bass developed the notion of a ‘‘Morita context,’’ which greatly facilitates the applications of the theory. Questions relating to the equivalence of categories through the use of Morita contexts have been the subject of many investigations. We refer the reader towx 6 for an extensive list of references. U. Knauerwx 9 and B. Banaschewski wx 1 , independently, transferred the ring theoretic approach to monoids. Knauer considered the category S-Act of sets over a monoid S. The main theorem ofwx 9 states that for monoids S and R, the categories S-Act and R-Act are equivalent if and only if there exists an idempotent e in S such that S s SeS and R ( eSe. Banaschewski made the important observation that if R and S are semigroupsŽ monoids without identity. , and the categories R-Act and S-Act are equivalent, then R and S are isomorphic semigroups. Inwx 14 we generalised Morita theory to semigroups having ‘‘local units,’’ where a semigroup S is said to have local units if for each x in S there exist idempotents e and f such that x s exf. We managed to show that a semigroup S with local units defines an endofunctor on the category S-Act and that this functor ‘‘fixes’’ certain objects in S-Act, that is, there exist M in S-Act such that the image of M is isomorphic to M. We were thus able to define the category S-FxAct of fixed objects. We defined two semi- groups R and S with local units to be Morita equivalent if the categories R-FxAct and S-FxAct are equivalent. We adapted the definition of a Morita contextŽ. see Section 3 for our purposes and then proved the following theorem.
Two semigroups S and T with local units are Morita equi¨alent if and only if ²²:: there exists a Morita context R, S, RSSRP , Q ,,,,,wx such thatR P g R- ²: FxAct, S Q g S-FxAct, and the maps ,,,wxare surjecti¨e. Moreover, we were able to give an analogue of the above theorem on simple Artinian rings. That is, a regular semigroup S is completely zero- simple if and only if it is Morita equivalent to a group with zero attached. By relying on some results of Hotzelwx 7 we were also able to recover the Rees theorem. This result was a direct consequence of the existence of a Morita context. In this paper, we use the notion of a Morita context to prove the 2 sufficiency part of the above theorem for semigroups of the form S s S . In fact, we find that a Morita context with certain additional properties allows us to define an equivalence relation, which we call strong Morita 2 equivalence, on the class of all semigroups of the form S s S . Our main theorem states that if R and S are strongly Morita equivalent then the STRONG MORITA EQUIVALENCE 373 respective fixed object categories, that is, R-FxAct and S-FxAct are also equivalent. This theorem can be viewed as the semigroup analogue of the celebrated theorem for rings which has come to be known as Morita Iwx 8 . We mention that inwx 6 Garcia and Simon develop a theory of Morita 2 equivalence for rings of the form R s R ; however, our methods appear to be very different from theirs. We also show that the category S-FxAct contains some important objects, namely, the indecomposable projectives. It turns out that in a large class of semigroups the indecomposable projectives are precisely those objects which are isomorphic to idempotent generated principal left ideals of the semigroup. This fact allows us to investigate properties of semi- groups which are governed by the idempotents. We show that we can construct Morita equivalent semigroups with ease. Our construction is based on the classical Rees theorem and a generalisa- tion of this theorem by Hotzelwx 7 . We then use properties of equivalent categories to deduce properties of this construction and to give a generali- sation of the Rees theorem. As applications we give a new proof of the Rees theoremŽ without relying on Hotzel’s results. . We also show that a given semigroup contains an abundant supply of strongly Morita equivalent subsemigroups. Finally, we apply our generalisation of the Rees theorem to the importantŽ from the viewpoint of group complexitywx 5 of semigroups. class of unambiguous semigroups.
2. PRELIMINARIES
For any undefined terms we refer the reader towx 5 or w 11 x . Throughout, we shall let ESŽ.denote the set of idempotents of a semigroup S. For our exposition we shall need the following types of semigroups:
2 I. A semigroup S is said to be factorisable if S s S , that is, every element of S can be written as the productŽ. not necessarily unique of two elements. II. We say that a semigroup S is a sandwich semigroup if there exists a set E of idempotents in S such that S s SES. III. A semigroup S is said to have local units if for each x in S there exist idempotents e and f in ESŽ.such that x s exf.
It is easy to see that a semigroup with local units is a sandwich semigroup and that a sandwich semigroup is factorisable. 374 S. TALWAR
2.1. The Category of Actions We begin by recapitulating some basic definitions fromwx 14 , referring the reader towx 14 for details.
DEFINITION 1. Let S be a semigroup. By an S action we mean a set M together with a function S = M ª M, denoted Ž.s, m ¬ sm, satisfying Ž.st m s stm Ž ..IfSis a monoid with identity 1, then we further require that 1m s m. We shall refer to M as an S-act. Let M and N be two S-acts. A mapping f: M ª N is called aŽ. left S-morphism if Ž.sm f s smf Ž.. The left S-acts together with the S- morphisms form a category which we shall denote by S-Act. As usual we shall denote a left S-act M by S M and the set of left S-morphisms from Ž. SSMinto N by Hom SSSM, N .If MsSand n is an element of N, then by Ž.n we denote the left S-morphism s ¬ sn, from S to N. Analogously, right S-acts are defined. Their category will be denoted by
Act-S. We shall distinguish left and right S-acts by writing SSM and M , respectively. We shall denote the set of right S-morphisms from MS into NSSSSby Hom Ž.M , N . However, we warn that we shall write right mor- phisms to the left of their argument. For a semigroup with 0Ž. zero we are concerned with pointed S-acts, where an S-act M is said to be pointed if it has a distinguished element, also denoted by 0, such that 0m s 0 for all m g M. We note that for this element 0 in M we have s0 s 0 for all s g S. The category of pointed S-acts will also be denoted by S-Act. Ä4 For a family Pii: i g I of S-acts, the coproduct @ gIiP exists and is isomorphic to the disjoint union of the sets Pii.IfShas a zero and the P are pointed S-acts, then their co-product is the 0-direct union. If a set M is simultaneously a left S-act and a right R-act such that Ž.Ž. smr s sm r, then we shall call M an S᎐R-biact and denote it by SRM .If SRNis some other S᎐R-biact, then we say that a mapping f: M ª N is an S᎐R-bimorphism if it is simultaneously a left S-morphism and a right R-morphism.
2.2. The Hom and Tensor Functors Given semigroups R and S, our aim is to seek conditions for certain subcategories of R-Act and S-Act to be equivalent. We shall need two important functors, the first of which is derived from the Hom set.
PROPOSITION 114.wx Let R, S, and T be semigroups andSR N and ST U be bi-acts. The set Hom SSŽ.U, SN becomes a left T-act and a right R-act if for Ž. ␣gHom SSU, SN , r g R, and t g T, we define the respecti¨e actions by
tи␣:ugU¬Ž.ut ␣ g N and ␣ и r : u g U ¬ Ž.u␣ r g N. STRONG MORITA EQUIVALENCE 375
For a fixed STU we obtain the functor Hom SŽ.U, from S-Act to T-Act, Ž. which is induced by the mappings N ¬ Hom S U, N and f g Ž. Ž. Ž. Hom SSM, N ¬ fЈ, where fЈ: Hom U, M ª Hom SU, N is defined by g¬gf. We note that if PTTis a right T-act, then the set Hom Ž.UT, PTof right T-morphisms becomes a right S-act if we define ␣ и s: u g U ¬ ␣Ž.su g P. Second, for a right S-act MSSand a left S-act N, M mSN is the solution of the usual universal problem, that is,
M mSN s Ž.M = N r , where is the equivalence relation on M = N generated by
⌺ s Ä4Ž.Ž.Ž.ms, n , m, sn : m g M, n g N, s g S . Ž. We denote the -class of m, n by m m n. As usual we refer to M mS N as the tensor product of M and N. It is easy to see that if M is an
R᎐S-biact then RSM m N is a left R-act. Moreover, for a fixed R᎐S-biact RSMwe may define the functor RSM m from S-Act to R-Act, which is induced by the mappings SRSN ¬ M m N, and any left S-morphism g: SN ªSMPmaps to I m g, where I Mis the identity map on M.
2.3. The Category of Fixed Actions Just as inwx 14 , we now proceed to pick an important subcategory of S-Act. We say that an S-act M is unitary if SM s M, where SM s Ä sm : s g S, m g M4. The unitary S-acts together with the usual S-morphisms form a full subcategory of S-Act, which we shall denote by S-UAct. We note that we can always form the subact SM of M. In particular if S is a factorisable semigroup, then SM is a unitary subact of M. Henceforth, all semigroups will be factorisable, unless otherwise stated. We define the functorial co¨er of S to be the semigroup with underlying set
S mSSS Hom Ž.S, S and multiplication
Ž.Ž.Ž.s m t и ␣ u m ¨ и  s stи␣mu¨и. It is easy to verify that the multiplication is well defined and associative. Moreover the map
⌫: s m t и ␣ ¬ Ž.st ␣ 376 S. TALWAR is a surjective semigroup homomorphism. If S is a semigroup with local units then by Corollary 4.4 ofwx 14 , ⌫ is an isomorphism. DEFINITION 2. We say that M g S-UAct is a fixed action if the S- Ž. Ž. morphism ⌫MS: S m S Hom SS, M ª M defined by s m t и ␣ ¬ st ␣ is an isomorphism. The class of fixed actions together with the usual left S-morphisms defines a category which we will denote by S-FxAct. We note that if S has a zero then 0 is always a fixed object.
DEFINITION 3. Two semigroups R and S are said to be Morita equi¨a- lent if the categories R-FxAct and S-FxAct are equivalent. We have designed the above definitions with the following in mind: Ž. LEMMA 1. The restriction of the functor S mSSS Hom S, to S-FxAct is naturally isomorphic to the identity functor on S-FxAct. Proof. Follows from the definitions. We may similarly define the category FxAct-S to be those unitary right Ž. S-acts which are fixed by the functor Hom SSS , S m SS. The following lemma will play a crucial role in our exposition.
LEMMA 2. If M is an object of S-UAct then S mS M is an object of S-FxAct.
Proof. Let s m m g S mS M and let m s khn for some k, h g S and n g M. Define
⌳: S mSSSM ª S m S Hom Ž.S, S m M by s m m ¬ s m k и Ž.h m n .
We note first that if m s kЈhЈnЈ then letting s s abc we have
s m k и Ž.h m n s abc m k и Ž.h m n sambиŽ.cmm sambиŽ.cmkЈhЈnЈ ssmkЈиŽ.hЈmnЈ .
It is now easy to see that the map ⌳ is well defined and a left S-morphism. Since the map ⌫ is well defined ⌳ is injective. SmS M STRONG MORITA EQUIVALENCE 377
Ž. To show that ⌳ is surjective let s m t и ␣ g S mSSS Hom S, S m M . Since t␣ s a m q for some a g S and q g M we have that stŽ.␣ ssa m q. If q s u¨p, then
Ž.sa m q ⌳ s sa m u и Ž.¨ m p ssmaиŽ.u¨mp ssmtи␣.
Thus ⌳ is an isomorphism and is the inverse of ⌫ . SmS M
COROLLARY 1. S mS is a functor from S-UAct to S-FxAct. Proof. The proof follows from our earlier comments on tensor prod- ucts. Ž. COROLLARY 2. S mSSS Hom S, S is an object of S-FxAct.
Proof. S Hom SŽ.S, S is an object of S-UAct. We note that the above definition of S-FxAct is not different from the one given inwx 14 . However, inwx 14 we were able to give a different characterisation of S-FxAct.
LEMMA 314.wx Let S be a semigroup with local units. A unitary S-act M is in S-FxAct if and only if S mS M ( M. In particular the counterpart to the main theorem of this paperŽ Theo- rem 3. was considerably easier to prove due to this characterisation.
3. PROJECTIVES
The category S-FxAct contains some important objects.
DEFINITION 4. P g S-Act is called projecti¨e if given any diagram
P f6 6 MNp of S-acts and S-morphisms with p surjective there exists an S-morphism g: P ª M such that the triangle is commutative. An S-act is said to be indecomposable if it cannot be written as the coproduct of two non-trivial S-acts. 378 S. TALWAR
For a semigroup with local units we were able to adapt the results ofwx 9 , and totally determine the nature of projectives.
THEOREM 114.wx Let S be a semigroup with local units. A unitary S-act P is projecti¨e and indecomposable if and only if P ( Se for some idempotent e in S. Moreo¨er, a unitary S-act P is projecti¨e if and only if P ( @ig IiP for some set I, where for each i g I there exists an idempotent ei of S such that Pii( Se . Furthermore, all projecti¨es in S-UAct lie in S-FxAct. It is easy to see that if S is any semigroup and e is an idempotent in S, then the left S-act, Se, is indecomposable and projective. Moreover, if e and f are idempotents in S then Hom SŽ.Se, Sf and eSf are isomorphic as eSe᎐fSf-biacts. It is also well known that in any category the coproduct of a family of projectives is a projective. We are not able to determine the structure of every projective for an arbitrary semigroup. Nevertheless, we have the following result.
PROPOSITION 2. Let S be a sandwich semigroup. A unitary S-act P is indecomposable and projecti¨e if and only if P ( Se for some idempotent e in S.
Proof. The proof is similar to the case when S is a semigroup with local unitsŽ seewx 14. .
DEFINITION 5. Let S be a factorisable semigroup. We say that P g S- UAct is a principal left projecti¨e if P ( @ig IiP for some set I where for each i, there is an idempotent eiiof S such that P ( Sei. Similarly we may define a principal right projecti¨e. We shall say that S is projecti¨e if S S is a principal left projective and SS is a principal right projective.
We mention that a principal projective is a projectiveŽ since each Sei is projective, and a coproduct of projectives is projective. .
PROPOSITION 3. Let S be a factorisable semigroup and let P be a principal projecti¨einS-UAct; then P is an object of S-FxAct. Proof. Let P be as described in the above definition; we claim that S mSSiP ( P. Define ⌽: S m P ª P by s m p ¬ spi, and for each i g I let iibe the isomorphism Se ª Pi. ⌽ is clearly surjective. To show it is Ž. Ž. injective, suppose spiis tq .If p is ue iii and q s ¨e ii for some u, ¨gS, then sueiis t¨e and it follows that sueiiiiiim e s t¨e m e , so that smpiistmq. It follows from the above that the principal projectives define a subcate- gory of S-FxAct. STRONG MORITA EQUIVALENCE 379
4. MORITA CONTEXTS AND AN EQUIVALENCE RELATION
The central concept of our machinery is the classical notion of a Morita context for modules over rings with identity, as presented inwx 8 . We drop the additivity clause and formulate the following:
DEFINITION 6. A six-tuple ²²:R, S, RSSRP , Q , , ,wx ,: is said to be a unitary Morita context, where R and S are factorisable semigroups, RSP and SRQ are respectively unitary R᎐S- and S᎐R-biacts,²: , is an R᎐R- morphism of P mSRQ into R, andwx , is an S᎐S-morphism of Q m P into Ssuch that the following hold:
1. ²:p, qpЈspqwx,pЈ 2. qp²:,qЈswxq,pqЈ.
We remind the reader that inwx 14 , we defined two semigroups S and T with local units to be Morita equivalent if the categories S-FxAct and T-FxAct are equivalent. We also proved the following theorem, which can be considered to be our motivating force.
THEOREM 214.wx Let S and T be semigroups with local units. Then S and T are Morita equi¨alent if and only if there exists a Morita context ²²:: R,S,RSSRP,Q,,,,,wx such thatR P g R-FxAct, SQ g S-FxAct, and the maps ²:,,,wxare surjecti¨e.
DEFINITION 7. We define two semigroups R and S to be strongly Morita equi¨alent if there exists a unitary Morita context such that²: , and wx, are surjective.
It is a routine matter to show that this is indeed an equivalence relation on the class of factorisable semigroups. A little surprisingly, just as in the case of rings with identity we have the following:
LEMMA 4. Let²²: R, S, RSSRP , Q ,,,,wx:²:be a unitary context with , surjecti¨e. If R is a semigroup with local units then the map ²:, from RSP m SQ R into R is also injecti¨e.
Proof. See Lemma 8.1 ofwx 14 .
We note that by the definition of a context a similar statement holds for S. 380 S. TALWAR
5. STRONG MORITA EQUIVALENCE IMPLIES MORITA EQUIVALENCE
We now proceed to prove the following theorem.
THEOREM 3. Let R and S be strongly Morita equi¨alent and let ²²::²: R,S,RSSRP,Q,,,,wx be a unitary Morita context with ,:PmSQªR and wx,:QmR PªS surjecti¨e. Then R and S are Morita equi¨alent ¨ia the functors
S mSRS Hom Ž.P,:R-FxActªS-FxAct and
RmRSRHom Ž.Q,:S-FxAct ª R-FxAct.
We shall prove the above theorem through a series of lemmas. ²: We note first that for each q g Q we may define the map, , q : R P ²: ²: ²: ªR Rby p ¬ p, q . Since , is a left R-morphism, it follows that , q is also a left R-morphism. Moreover, since q s sqЈ for some s g S and ²: ² : ²: Ž . qЈgQand , q s s и , qЈ , we have that , q g S Hom R P, R . Fur- thermore, for any n g N, where N g R-Act, the map²: , q и n is in Ž. SHom R P, N . Similarly for each p g P we may define the left S- morphism,wx , p : SSQ ª S by q ¬ wq, p x. LEMMA 5. Let R and S be strongly Morita equi¨alent. For M g S-FxAct and N g R-FxAct the maps
␣ : RSP m M ª R Hom SŽ.Q, M defined by p m m ¬ wx, p и m and
 : SRQ m N ª S Hom RŽ.P, N defined by q m n ¬ ²:, q и n are surjecti¨e left R- and S-morphisms, respec- ti¨ely. Proof. It is easy to see that ␣ and  are well defined. Since
Ž.rp m m ␣ s wx, rp и m s r и wx, p и m s rpŽ.mm␣ we have that ␣ is a left R-morphism. Similarly,  is a left S-morphism. Ž. To show that  is surjective, let s и g S Hom R P, N . Sincewx , is surjective there exist q g Q and p g P such that wxq, p s s, also p g N, and we have that Ž.²:q m ps ,q иp. Let pЈ g P. Then,
pЈ²:,qиps ²pЈ,q :иpspЈwxq,pspЈŽ.sи. STRONG MORITA EQUIVALENCE 381
We now write
< < SmSR< RmRS< LEMMA 6. < LEMMA 7. The isomorphisms in the following are natural SmSR< RmRS< Proof. Let f: RRN ª M bealeft R-morphism, and let Ž. ²: ⌽NR:< ⌽ 6 N Ž. < 6 Ј fЉ6 f 6 Ž. < s< I ⌽ 6 SNm Ž. SmSR< Љ IfЈ 6 ISmf6 Sm 6 Ž. S mSR< ⌸␣: RSP m S m SS Hom RŽ.P, N ª R Hom R Ž.R, N Ž. by p m s m t и ¬ pst . The map ⌸␣ is a surjecti¨e left R-morphism such that ker ⌸␣ s ker ␣. Ž. Proof. Let p, pЈ g P, s, sЈ, t, tЈ g S and , Ј g Hom R P, N .Itis Ž. Ž. easy to see that pst g R Hom R R, N . Now suppose that Ž.Ž.p m s m t и ␣s pЈmsЈmtЈиЈ ␣. Then for all q g Q we have wxq, p и Ž.s m t и s wxq, pЈиŽ.sЈmtЈиЈ, and so for all pЉ g P we have pЉ Žwxq, pstи.ŽspЉwxq,pЈsЈtЈиЈ., that is, Ž.Ž.²:pЉ,qpss ²:pЉ,qpЈsЈЈ. Since²: , is surjective, we may write any r g R as ²pЉ, q :. Thus Žpst . Ž. spЈsЈtЈЈ, and it follows that ker ␣ : ker ⌸␣ . Conversely suppose that Ž.Ž.p m s m t и ⌸␣s pЈ m sЈ m tЈиЈ⌸␣. Now Ž.p m s m t и ␣swx,psŽ.mtи:q¬wxq,psmtиand ŽpЈ m sЈ mtЈиЈ.␣swx, pЈŽ.sЈmtЈиЈ: q¬wxq, pЈ sЈmtЈиЈ. If we can show that wxq, pstиs wq,pЈ xsЈtЈиЈ for all q g Q, then writing q s uqЈ for some u g S and qЈ g Q, we will have umwxqЈ,pstиsum wqЈ,pЈ xsЈtЈиЈ and the equality ker ␣ s ker ⌸␣ will follow. Let pЉ g P and q g Q; then pЉŽwxq,pstŽ.иs²:pЉ,q Ž. pst s ²:pЉ , qp ŽЈsЈtЈ .Ј spЉŽ.wxq,pЈsЈtЈЈ. Thus wxq, psŽ.mtиswxq,pЈŽ.sЈmtЈиЈ. It only remains to show that ⌸␣ is surjective. Let r и g R Hom R Ž.R,Nand let r s st for some s, t g R. Now s s ²:p, q for some p g P ²:Ž. Ž . and q g Q; also, , q и t g S Hom R P, N . Writing p s pЈu for some pЈ g P and u g R, we have that pЈ m u m ²:Ž., q и t is mapped to r и . STRONG MORITA EQUIVALENCE 383 Similarly, for M g S-UAct, we may define the map ⌸: SRQ m R m RR Hom SŽ.Q, M ª S Hom S Ž.S, M Ž. by q m r m rЈи¬ qrrЈ and show that ⌸ is a surjective left S- morphism such that ker ⌸ s ker . As before, we write << PmSSSmSHom RŽ.P, N ⌸␣ for Ž.P mSSS m S Hom RŽ.P, N rker ⌸␣ and << QmRRRmRHom SŽ.Q, M ⌸ for Ž.Q mRRR m R Hom SŽ.Q, M rker ⌸. For N g R-FxAct, the following holds. HGNŽ.Ž.(RmRSS< (RmRRRHom Ž.R, N , by Lemma 8 (N, by definition of R-FxAct, with all isomorphisms natural. Similarly, for M g S-FxAct GHMŽ.Ž.(SmSRR< (SmSSSHom Ž.S, M (M. The proof of Theorem 3 is now complete. It is also possible to recover a great deal of information from a context. The following theorem can be viewed as the semigroup analogue of the celebrated theorem for rings which has come to be known as Morita Iwx 8 . THEOREM 4. Let R and S be strongly Morita equi¨alent and let ²²:R,S,RSSRP,Q,,,,wx:²:be a unitary Morita context with , and wx, surjecti¨e. Then the following hold: 1. The categories R-FxAct and S-FxAct are equi¨alent. Ž. Ž. 2. R m R Hom RRSSR, R ( R m R End Q and S m S Hom S, S ( SmSRSEnd P as semigroups. 384 S. TALWAR Ž. 3. IfRS R g R-FxAct, then S m S Hom RP, R ( S mSS Ž. Ž. Hom SSS, Q and if S g S-FxAct, then R mRR Hom SRQ, S ( R m RHom RŽ.R, P as left acts. Ä4 4. Let LR denote the lattice I : R : I is an ideal of R, RIR s I and similarly for LSR. Then there is a lattice isomorphism between L and LS. Proof. We have already provedŽ. 1 . 2. Let  be defined as in Lemma 5 and define Ј: Q mRSP ª S Hom Ž.S, S by qmp¬Ž.wxq,p. We claim that ker  s ker Ј. Suppose that²: , qps ²,qЈ :pЈfor some q, qЈgQand p, pЈ g P. Let s s wxqЉ, pЉ be an arbitrary element of S. Thus sŽ.wxwq,psqЉ,pЉ xwxq,pswxqЉ,²:pЉ,qpssŽ.wqЈ,pЈ x. Conversely, let Žwxq, p .Žs wxqЈ, pЈ ., and for all pЉ g P we have pЉwxq,pspЉ wqЈ,pЈ x. Thus ²:²:pЉ, qps pЉ,qЈpЈ, and the claim follows. By Lemma 7 we have SmSRSEnd P ( S m SR< Ž. 3. We note that if R g R-FxAct then R ( R m R Hom R R, R and Ž. so by 2 , R ( R mRSR End Q. Thus, SmSRSHom Ž.P, R ( S m SR< (SmSRR< (SmSSSHom Ž.S, Q by Lemma 8. Ä 4. Let I g LR and let wxQI, P denote the set wxqa, p : q g Q, a g I,pgP4. It is easy to see that wxQI, P is an ideal in S; also, SQIwx,PSs wxwxSQI, PS s QI, P . ²: IJ: 4 For J g LS let PJ, Q denote the set pb, q : q g Q, b g J, p g P . ²: Again it is easy to see that PJ, Q g LS. STRONG MORITA EQUIVALENCE 385 ²: Define maps LRSª L by I ¬ wxQI, P and LSRª L by J ¬ PJ, Q . Thus, I¬wxQI, P ¬ ²:²²::²:²:PQI wx,P,Q s P,QI P, Q s P, QIP,Q sI. The two maps clearly preserve inclusion and the result follows. COROLLARY 3. If R and S be strongly Morita equi¨alent semigroups, then the categories FxAct-S and FxAct-R are also equi¨alent. Moreo¨er, the dual statements of the abo¨e theorem hold. Proof. Follows from the symmetric nature of a Morita context. We have shown that strong Morita equivalence implies Morita equiva- lence. We shall make free use of the following well known facts on equivalent categories. Let C and D be equivalent categories via the functors F: C ª D and G: D ª C; then Ž. I. If M, N g C and HomC M, N denotes the set of morphisms in Cfrom M to N, then there is a bijection of the sets HomC Ž.M, N and Hom DCŽŽFM .,FN Ž ...InparticularHomŽ.M,M and Hom D ŽŽFM .,FM Ž ..are isomorphic as semigroups. II. A morphism f in C is an epimorphismŽ. monomorphism if and only if FfŽ.is an epimorphism Ž monomorphism . in D. Ä4 III. If MiigIiis a family of objects in C such that @ gIiM is an Ž. Ž. object of C, then F @ig IiM is isomorphic to @igIiFM in D. IV. I is an indecomposable in C if and only if FIŽ.is an indecom- posable in D, where an S-act is said to be indecomposable if it cannot be written as the co-product of two non-empty acts. V. P is a projective in C if and only if FPŽ.is a projective in D. As immediate applications we have: COROLLARY 4. Let R and S be sandwich semigroups. If R and S are strongly Morita equi¨alent, then the subcategories of principal projecti¨es are equi¨alent. In particular, the subcategories of indecomposable projecti¨es are equi¨alent. Moreo¨er, if e and f are idempotents in R then there exist idempotents eЈ, fЈ in S such that eRf and eЈSfЈ are isomorphic as semigroups. Proof. The proof follows from the earlier facts on equivalence functors and Proposition 2. COROLLARY 5. Let R and S be sandwich semigroups. If R and S are strongly Morita equi¨alent then the cardinalities of the sets of regular D-classes of R and S are the same. 386 S. TALWAR Proof. It is well known that in any semigroup S the idempotents e and f are D-related if and only if Se ( Sf as left S-acts and that a D-class is regular if and only if it contains an idempotent. We choose one idempo- tent from each regular D-class of R and denote this set by ER; similarly we may choose one idempotent from each regular D-class of S and denote this set by ES. That the cardinalities of these two sets are the same follows from the above facts on equivalence functors and Propositions 2 and 3. 6. A CONSTRUCTION We now show that we can construct strongly Morita equivalent semi- groups with ease. Our construction is an analogue of the following impor- tant classical construction as presented inwx 5 . Let R be a semigroup, let X and Y be finite non-empty sets, and let ²:,:Y=XªR be a functionŽ²:. with values denoted y, x . Then the set M s X = R = Y equipped with the associative product Ž.Ž.x,s,yxЈ,sЈ,yЈsŽ.x,sy²:,xЈsЈ,yЈ is a semigroup, known as the Rees matrix semigroup o¨er R defined by ²:,. We formulate the following: Let R be a semigroup, and let RRP and Q be respectively left and right R-acts. Also, let ²: ,:RRP=QªR be an R᎐R-bi-linear, that is, ²rp, q :²:s rp,q and ²p, qs :²:s p, qs. Then the set Q mR P becomes a semigroup with product Ž.Ž.qmpqЈmpЈsqm²:p,qЈpЈ. The multiplication is well defined since²: , is an R᎐R-bi-linear map. It is easy to verify that it is associative. We shall refer to Q mR P as the Morita semigroup o¨er R defined by ²:,. We mention that if R is a monoid with zero, then the data in the above define a dual R-operand in the terminology ofwx 7 . THEOREM 5. Let R be a factorisable semigroup, and letRR P and Q be respecti¨ely unitary left and right R-acts. Also, let ²: ,:RRP=QªR STRONG MORITA EQUIVALENCE 387 be a surjecti¨eR᎐R-bilinear map. The Morita semigroup Q mR P is strongly Morita equi¨alent to R. Ž Proof. First, we show that Q mRRP is a semigroup of the form Q m .2 PsQmRP. Let q m p g Q mRRP. Since P is unitary there exist r g R, pЈ g P such that p s rpЈ. Also, since²: , is surjective there exist pЉ g P, qЉ g Q such that ²:pЉ, qЉ s r, and it follows that Ž.Ž.qmpЉqЉmpЈsqm²:pЉ,qЉpЈsqmp. Ž. Pbecomes a unitary R- Q mR P -biact if we define a right action by pиŽ.qЈmpЈs²:p,qЈpЈ, where p g P and qЈ m pЈ g Q m P. That the action is well defined follows from the bilinearity of²: , . Ž. Similarly, Q becomes a unitary Q mR P -R-biact if we define a left action by Ž.qЈmpЈиqsqЈ²:pЈ,q. It is now easy to verify that ²²:: R,QmRP,P,Q,,,,wx is a Morita context, wherewx , is the identity map. The maps²: , andwx , are surjective by construction. A Morita context also gives rise to new semigroups. PROPOSITION 4. Let²²: R, S, RSSRP , Q ,,,,wx:be a unitary Morita con- text and let p, pЈ g P and q, qЈ g Q. The bi-act P mS Q is a semigroup with Ž.Ž. multiplication p m qpЈmqЈspqwx,pЈmqЈand the bi-act Q mR Pisa semigroup with multiplicationŽ.Ž.²: q m pqЈmpЈsqp,qЈmpЈ.Also, the maps ²:, and wx, are semigroup homomorphisms. If the two maps are surjecti¨e then the semigroups Q mRSP, P m Q, R, and S are all strongly Morita equi¨alent. Proof. The associativity of operations follows directly from the defini- tions of²: , andwx , . COROLLARY 6. Let R be a semigroup with local units, and letRR P and Q ²: be respecti¨ely unitary left and right R-acts. If ,:RRP=QªRisa surjecti¨eR᎐R-bilinear map, then Q mR P is a sandwich semigroup. Proof. We show first that each idempotent e g R gives rise to an idempotent in Q mR P. Let e be an idempotent in R; then there exist 388 S. TALWAR pЈgPand qЈ g Q such that ²:pЈ, qЈ s e. In particular ²epЈ, qЈ :s e and it follows that Ž.Ž.qЈmepЈ qЈ m epЈ s qЈ m ²:epЈ, qЈ epЈ s Ž.qЈ m epЈ . ÄŽ.²:4 We let E s qЈ m epЈ : pЈ, qЈ s e . Now, let q m p g Q mR P. Since R is a semigroup with local units and R P is unitary there exists an idempo- tent e in R such that ep s p. Let ²:pЈ, qЈ s e, then by the above Ž. qЈmepЈ is an idempotent of Q mR P and Ž.Ž.Ž.Ž.q m epЈ qЈ m epЈ qЈ m ep s q m p . 7. CO-ORDINATES The Morita semigroup can be given a special form in the following situation. ²: Let Q mR P be a Morita semigroup over R defined by , . Suppose moreover that RRP is a principal left projective and Q is a principal right projective. So that, for some set ⌳, R P ( @g ⌳ P , Ž. where for each there is an e in ER such that P ( Re, and for some set I, QRi( @ gIiQ , Ž. where for each i there is an e in ER such that Qi ( eR. We now choose an idempotent from each regular D-class of R and Ä4 denote this set by E. Let EP s e g E : ᭚ g ⌳, P ( Re . Thus EP con- tains those idempotents of E which are required in the decomposition of Ä4 P; also, for a fixed e g EPelet ⌳ s g ⌳ : P( Re . We shall denote an element of ⌳ eeby . Then, P P, where P Re for each e E . R (@@ ee( g P egEPeg⌳ e If denotes a left R-isomorphism Re P , then each p P can be e ª e g written as Ž.se for some s R, e E and ⌳ . e g g eeg Ä4Ä4 Similarly, let EQis e g E : ᭚i g I, Q ( eR and Ies i g I : Qi( eR . Then denoting an element of Ieeby i , we have Q Q, where Q eR for each e E . Rii(@@ ee( g Q egEiQegI e STRONG MORITA EQUIVALENCE 389 Again let denote a right R-isomorphism fR Q . Then writing a iif ª f right morphism on the left, we have that each q g Q can be written as Ž.ft for some t R, f E, and i I . iffg g g f If q m p s qЈ m pЈ is an element of Q mR P, then there exists a se- quence Ž.Žq, p s q11, p .Žª q 22, p .ª иии ª Žqnn, p .Žs qЈ, pЈ . ŽŽ . Ž .. ⌺ such that for each k s 1,...,n either qkk, p , q kq1, pkq1g or ŽŽ . Ž ..⌺ Ž . qkq1, pkq1 , qkk, p g see Section 2.2 . ŽŽ . Ž .. ⌺ Suppose qkq1, pkq1 , qkk, p g ; then there exists ¨ kg R, such that qkksqq1¨kkand p q1s ¨ kkp . It follows that there exist uniquely defined e g EPQff, f g E , i g I , and ⌳, such that q ,...,q Q , and p ,..., p P . eeg 1 nig fe1 ng Put I I and ⌳ ⌳ and let q p be an element of s Deg EeQPs DegEe m QmP. By the above, Ž.q, p ft, se . s Ž.ife Ž. If q m p s qЈ m pЈ, then for each k s 1,...,n we may write qkk, p s ŽŽ.Ž..ft , se . Now suppose ikfe k Ž.Ž.ft , se , Žft .Ž, se . ⌺. ž/Ž.ikfe k Ž.ikfq1 kq1 eg Then there exists R such that Ž.ft Žft .and Žse . ¨ kig ffks ikq1¨ kkq1e Ž.se and it follows that ft ft and se se, so that s¨kk e kks q1¨ kkq1s¨kk ftkk s e s ft kq1¨ kksesft kq1 skq1 e. This equation also holds if Ž.Ž.ft , se, Ž.Ž.ft , se ⌺. Ž.ikfeq1 kq1/žŽ.ikfe k g Thus q p f freŽ. e m s Ž.ifem for some unique fre g EREQP. Ž. We may thus define a map from Q mR P to the set of I = ⌳ matrices with at most one non-zero entry from EREQPby qmp¬Ž.fre ife,. 390 S. TALWAR We have shown that the map is well defined. It is clearly injective. We note that if R is a monoid and the acts P and Q are free, that is, each P is isomorphic to R as a left R-act and each Qi is isomorphic to R as a right R-act, then the map is also surjective. Moreover, a bilinear mapping ²:,: P Q R @@= iª can be represented by the Ž.⌳ = I -sandwich matrix X²:e,f. sefi DEFINITION 8. Let Q mR P be a Morita semigroup over R defined by ²: , . We say that Q mRRP has tensorial co-ordinate form if P is a principal left projective and QR is a principal right projective. We have established: THEOREM 6. If a Morita semigroup Q mR P has tensorial co-ordinate form then it can be embedded in the Rees matrix semigroup o¨er R, consisting of<< I = <⌳ THEOREM 7. Let R and S be strongly Morita equi¨alent with S a projecti¨e semigroup and R a semigroup with local units. Let²²: R, S, RSSRP , Q ,,,,wx: be a unitary Morita context with ²:,,,wxsurjecti¨e. Moreo¨er, suppose that RRP g R-FxAct and Q g FxAct-R. The semigroup Q mRP has tensorial co-ordinate form. Proof. Let S S ( @g ⌳ S and SSi( @ gIiS , where S( Se and Ž. Sii(fSfor some e, fiSg ES. Then by Corollary 3, S g S-FxAct and by part 3 of Theorem 4 RRSP ( R m R Hom Ž.Q, S . Now, F RRHom Ž.Q, is an equivalence functor, so that s mRS PFSŽ. FS FSŽ.. RS((ž/@@( g⌳ g⌳ STRONG MORITA EQUIVALENCE 391 Since each S is an indecomposable projective we have that FSŽ. is also an indecomposable projective in R-FxAct, and it follows that there exists Ž. Ž . egER such that FS (Re. We may give a similar description of QR , and we are thus in the situation described above. 8. APPLICATIONS 8.1. The Rees Theorem Our first application is a new proof of the classical Rees theoremwx 12 , for completely 0-simple semigroups. Let S be a completely 0-simple semigroup with nL-classes and mR-classes. We shall take the following facts for granted. I. S is regular. II. There exists an idempotent e in S, such that S s SeS. III. S is a 0-direct union of n left ideals isomorphic with Se as left S-acts. IV. S is a 0-direct union of m right ideals isomorphic with eS as right S-acts. 0 V. eSe s G is a group with an attached zero. Thus, S gives rise to a Morita context ²S,G0 ,Se, eS,,,,²:wx : , ²: 0 where , : eS mS Se ª G is defined by es m te ¬ este andwx , : Se meSe eS ª SeS is defined by se m et ¬ set. It is easy to check that the hypotheses of the above theorem hold. If 0 F: S-FxAct ª G -FxAct denotes an equivalence functor then nn n FSŽ. FSe FSeŽ. Gx0, Ssž/@@( ( @ s1 s1 s1 0 0 0 where each G x is isomorphic to G as a left G -act. The last isomor- 0 phism follows from the fact that FSeŽ. is a projective in G -FxAct, and the only non-zero idempotent in G0 is the identity of G. Thus, se m et s Ž.ie m ge Ž., which can now be described as an I = ⌳ matrix with all entries zero except for g in the Ž.i, position. Also, the bilinear map ²: , can be described as an m = n matrix with entries in G0. 392 S. TALWAR 8.2. The Depth View of a Finite Semigroup Morita equivalent semigroups are abundant within a given semigroup. Let T be an arbitrary semigroup and let E be some set of idempotents in T. Put S s TET s SES and R s ETE s ESE. Then clearly S is a sand- wich semigroup and the semigroups S, R are Morita equivalent via the context ²²::S,ESE, SE, ES,,,,wx wx,:SE mESE ES ª S, where se m ft ¬ seft ²: ,:ES mS SE ª ESE, where es m tf ¬ estf. Let S be a finite semigroup, and suppose that the maximal D-classes of S are all regular. We choose an idempotent from each maximal D-class of S and denote this set by E11. Then S s SE S, so that S is strongly Morita equivalent to ESE11. It is well known that the maximal D-classes of ESE11are all groups. Thus we are developing a tool to relate a given semigroup to one with a simpler structure. We will find that we can learn about the original semigroup via the simpler one. Let Ž.G be the disjoint union of the maximal D-classes of D eegE1 ESE11. It is also well known that for a finite semigroup a maximal D-class is either regular or a non-idempotent singleton. Let C1 denote the collec- tion of all maximal non-regular D-classes of ESE Ž.G and put 11yD eegE1 SX ESE Ž.G C. 211s yDeegE1y 1 X If some maximal D-classes of S21are still non-regular, then we add to C all these non-regular maximal D-classes and again denote the resulting set by C1. Proceeding in this manner we have a semigroup S ESE Ž.G C 211s yDeegE1y 1 all of whose maximal D-classes are regular. We may thus repeat the above operation on S22and form the set E .It follows that for any finite semigroup S there exist subsemigroups S1,...,Sn such that Skks ESE1k1y Ž.GekeE yC1 yyD gky1y is strongly Morita equivalent to ESEkk. Moreover, Skis an ideal in ESEky1 ky1. In particular if S is a regular semigroup then Ck is empty for all k. STRONG MORITA EQUIVALENCE 393 8.3. Unambiguous Semigroups Birgetwx 3 introduced the notion of unambiguous semigroups. DEFINITION 9wx 3 . An order F on a semigroup is said to be unambigu- ous if for all x, y, z g S _ Ä40, zFx and z F y imply either x F y or y F x. A semigroup S is said to be L-unambiguous if the L-order of S is unambiguous. Dually we may define R-unambiguous. If both the R- and L-orders of S are unambiguous then we call S an unambiguous semigroup. Inwx 3 Birget also gives the following example which is credited to Rhodes. Let S be a finite semigroup and let S L denote the left Rhodes expansionŽ seewx 5 for the definition.Ž of S and S L.Rdenote the right Rhodes expansion of S L. THEOREM 8wx 3Ž. Rhodes . For any finite semigroup S the semigroup ŽSL .R is a finite unambiguous semigroup and has the same group complexity as S. In particular S is regular if and only ifŽ SL .R is regular. Birgetwx 3 also showed that if E is some set of idempotents in an unambiguous semigroup S then ESE is also an unambiguous semigroup. In particular any ideal of a regular unambiguous semigroup is also unam- biguous. THEOREM 9. If S is a finite regular unambiguous semigroup, and E denotes a set of idempotents formed by choosing one idempotent from each maximal D-class of S, then S ( SE mESE ES and the latter has tensorial co-ordinate form. Proof. It is easy to show that a regular unambiguous semigroup S is a projective semigroup and all other hypothesis of Theorem 7 hold. We mention that inwx 16 we show that every unambiguous semigroup is strongly Morita equivalent to a semigroup with tensorial co-ordinate form of the same group complexity. Inwx 15 we give a further application of the results of the present article. The starting point of the Synthesis Theorem as presented inwx 4, 13 is the concept of a structure matrix semigroup, inspired from Rees’ theorem. Next one extends such semigroups by groups. Finally, one iterates both operations, alternatingly, a finite number of times. Semigroups constructed in this manner have come to be known as iterative matrix semigroups Ž.IMS . In bothwx 13 and wx 4 it is shown that every semigroup divides a finite IMS in a ‘‘nice’’ way. The Morita semigroup is our analogue of a Rees structure matrix semigroup. Inwx 15 we formulate a new approach to extensions by groups and then rely on our results on equivalence of categories and our descrip- tion of projectives to give a new concise form of the Synthesis Theorem. 394 S. TALWAR ACKNOWLEDGMENTS The author is truly grateful to The Royal Society for funding during his stay in Paris, and indebted to J. B. Fountain, S. Margolis, P. Weil, and all at LITP for their kindness. REFERENCES 1. B. Banaschewski, Functors into the category of M-sets, Abh. Math. Sem. Uni¨. Hamburg 8 Ž.1972 , 49᎐64. 2. H. Bass, The Morita theorems, lecture notes, University of Oregon, 1962. 3. J. C. Birget, Iteration of expansionsᎏUnambiguous semigroups, J. Pure Appl. Algebra 34 Ž.1984 , 1᎐55. 4. J. C. Birget, The Synthesis Theorem for finite regular semigroups, and its generalization, J. Pure Appl. 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