.,i, . €{ A3,N,L'Mathai' Professor' u. :.,i;'.@ All rights reserved I:,.. ::.i:: .,_ , it,.r. i t i:j .1 r, ,:.,,.i:i i ::j1.. -: 'a .' '': :"::'.;' * 'lj r;jl4 ::\ ... - : . '-.:il-.1 Piiureition No.15 Centre for Mathematieal,:Sciences Vazhuthacad, Trivandrum 695014 I(erala State, India ' Phone: 65291 1989 STRUCTURE'Or REGULAR SEMIGROUPS' rr ' CROSS_CONNECTIONS by K.r.:S-;:s NAM B OORIPAD real', Canada Centre for-:\I at hematical Siiefi.c"s. Trivandrum. India Publication \o.15 Ceutre for }lathematical Sciences \;azli utlta.c,ad. Trivan drum 6950 14 Kerala State, India Pltorte:65291 . 1989 I PREFACE This book is intended as a sequel to the author's memoir published by the American Mathematical Soeiety in 1979. The author presented part of this mate- rial.in a talk in the confetence on 'Theory of Regular Semigroups and Applications' conducted in the Department as well as in the talk given at the semigroup theory _peuegu1g1 conducted along with ihe American ll'Iathematical Society Conference in i; the faifiof 1987 in Lincoln. Also a series of seminars were conducted in the Depart- ment of:.tittqgg.ratics, University of Kerala on this material. The author wishes to thank Mr. C. S. Ling of McGill University, Canada, a student of Professor A. M. Mathai for the painful task of computer-setting the manuscript, Mr. K. Gopinatha Panickcr and lr{iss K. T. Martiakutty of the Centre for Mathematical Sciences for looking after the printing and the Centre for Mathematical Sciences, Trivandrum- 14, Kerala State, India and its Director for bringing out this monograph in their publication series. K. S. S. Nambooripad f ,l * CONTENTS !.', Page $ a Preface ': 1 ...,..'.', ' 't' Introduction ; .r,,:.\ o ..li' t" b 1. Normal Categories : I 2. Projective'ComPlexes -: 18 :. i, ;" ;. :t.- ,, ;.: li., "1*+,,",Nq,6tal,DualS 30 4.', - . :,. '.'..:.Gfoss-counections i' 50 :t.I 5,,lVIorphism of CJoss-contiectiops' .i Iil , 65 I li d Se r n i g r o tt 1r o. tn" Cross-cenn eclion's ur ; nu;i{.1r1i} eU. ul ':,: ',: . i' il: . f .' :: . "inL i) :"= T 7. Remarks and Ex'3'iiffts . ' ;, ,,. '...,,, ., . .. Llr'' il:''.,i I 5t) R,q 91 I STRUCTURE Or REGULAR SEMIGR,OUPS, il CROSS-CONNECTIONS K. S. S. NAMBOORIPAD Introduction At present there are two established procedures for analysing the structure of an arbitrary regular semigroup-the theory of biordered sets (cf. [3] and [22]) and the theory of crss-conn'rctions (cf. [9] and [21]). The concept of a biordered set gives,a concrete foriifor the structure of the set of idempotents of a semigroup and thus'helps'to'visiralize this important structural component as a mathemat- ical olject on its.,gyl right. T.he.four spiral biordered set? [2] is an especially niceexample of.a;ffird.ered set,'w'trithican be visualized in this manner. Other interesting exarifples of biordered set$ can be found in [7]. We refer the reader to l22l fot the definition;:of biordere$,pe,ls. where it'is also proved that every regular biordered set corhes frcirn a rbeuiei..sp,p-1erOu-n. The result that every biordered set'comes frorna'semigroup i*a""li".p&dirwn(cf.l [o]). On the other hand, a cross-connection characterizes the idsal Structure of a regular semigroup. These "r*heories are complementary,in the sense that, while the theory of biordered sets ,' i6 more elementary and explicit, the theory of cross-connections is essentially ab- stract and so it is more natural when one studies semigroups that arise in diverse contexts, such as semigroups of matrices, linear operators, etc. The scope of these theories is limited by the fact that they mainly apply to study fundamental regular' semigroups. Recall that a semigroup S is fundamentalif the only congruence on 5 contained in the Green's relation lf is the identity congruence. However, the concept of. inductiue grcupoids, discussed in 1221, is useful in studying arbitrary regular semigroups. In this'paper we shall consider a generalization of Grillet's theory of cross-connections that will be complementary to the theory of inductive groupoids. Let ,5 be a regular semigroup. Hall [11] showed that there is a representation $ of S bypairsof mappingsonthepartiallyorderedsets.I = S/n and A = SlL, where f;. and A are Green's relations, defined as follows: for each r € ,S 6@): (r",1,) NAMBOORIPAD where r': A * Lll,z I ---+ 1] is the map defined by Lur, = Lu,fI,Ru = .&"]' He gave a showed that 6 is faithful if and only if 5 is fundamental. In [11]' Hall also construction of the maximum fundamental regular semigroup associated with '9' Grillet's theory of cross-connections may be viewed as a refinement of Hall's partially ideas. Grillet characterized the partially ordered sets f and A as regular A ordered sefs and the relations between them in terms of cross-connect'i'ons' -* and cross-connection between -I and A consists of two mappings I: -I Ao A:A --, Io of I and A intotworegularpartiallyorderedsets Ao and ro of regular (cf' equivalence relations on A and .I respectively satisfying Grillet's axigms [10])' this He showed that the maximum fundamental regular semigroup determined by the given data is the set of all pairs of normal mappings that are compatible with concept of cross- cross-connection (cf. [21], [26]). In order to distinguish Grillet's we connections from the more general concept considered in this Paper henceforth shall refer to the former as fundamentol crcss-conner'tions.. Now pairs of normal mappings compatible with a given cross-connection can the be viewed as an abstract description of the pairs of mapping that appear in the other hand, image of Hall's representation / of the regular semigroup sl;:on since ,5 is werr,kly reductive, the representation ry' defined'by: ,h@) = (P,,\,) eguivalent to where p,[]"] is the right [left] translation by c € .9 , is faithful and is In order to characterize the pairs Hall's representation / when ^9 is fundamental. we replace the partially ordeied sets .[ = Sl9 of -rppiogs (pr,)") abstractly, and A = SIL in Grillet's theory by small categories [,(S) and R(S) ' where whose objects are principal left ideals of ^9 [,(s) tR (s)] is the category {right] -and e whose morphisms are right [eft] translations. Thus p € llomsltl(Sc,5y) [l Homsls;(ts, y,S)] if and only if p: St -' Sy.[l' nS ' yS] is a mapping such that is regular for all'u, a € S, (ua)p = u(u)p[)(zu) : '\(u)u]' We show that when '9 if [,(s) and R.(5) are normal.red.ucti,ue categories (cf. Theoret 9.,81- conversely, all normal cones € is any normal, reductive category, then the semigroup ?€ of (cf. Theorems in 0 is a regular semigroup such that [,(?g) is isomorphism to € into the 3.3 and g.1i). Moreover in this case, there is an embedding of R(2.€) (cf. Theorem 3'14)' functor category €* = [€, set] of all set valued functors on € and The image N*€ of this embedding is a normal reductive subcate$ory of e* is called the normal dual of e. STIIUCTUttE OF REGULAR SBMIGROUPS Let € and D be two normal reductive categories. A cross-connection between *€ e and D is a local isomorphism f : D --' .lf such that the image of f is total in N.€ (cf. Definition 4.2). Notice that a cross-connection is a category equivalence (cf. [28], p.55). Furthermore, there is a local isomorphism f*:e -' N*D and a natural isomorphism Xr:f(-, -) - I*(-, -), where f(-'-) denotes tlre bifunctor associated with | (cf. Theorem 4.5). If c e ae and d € oD , then f(c,d) isasetof normalconesin € withvertex c and f.(c,d) isasetofconesin D with vcrtex d. We say that the pair (p,)), p € f(c,d), ) € l.(c,d),,is linked if 1.(c, d)(p) = ). The set of all linked pairs of normal cones, under a suitable rnultiplicatiori is a regular semigroup ^9f such that [,(Sf ) is isorrtorphic to € and R(Sf) is isomorphic to D. Every regular semigroup is up to isomorphism of this forrn (cf. Theorems 4.i3 and 1.15). Let l: D - N'€ and f': D' -- N'Qt be two cross-connections. A morphism rrr:l - I' is apair nz = (a, G) rvhere F:e - €' and G:D'- D' are(inclusion- preserving) functors satisfying some compatibility conditions (cf. Definition 5.1). \Ve prove that a morphisnt m:l -- I' induces a homomorphisrn -c;,r:5f - ^9I' and tha,t the assignment .9: I r* .9f **, sm is an equivalence of the catcgory of regular semigroups with the category of cross- connections. In $6 we investigate the relation between the concept of cross- connections introduced in this paper and that of fundarnental cross-connections (of Grillet). \Ve remark that the theory of cross-connections presented here provides a u1if;cd frarnework for studying various classes of regular semigroups. The use of abstract categories to formulate the theory is particularly useful in tliis context. Thus. for e,xample, to study sernigroups of ma.trices, it is natural to interpret the ruorrnal categories involved, as categories of vector spaces and it can be seen that such interpretation would lead to a natural interpretation of the dual category a1d the cross-colulection in terms of the vector space dual and the bilinear form associated with the dual. An examiuation of the axioms for normal categories would show that na,,ural examples of such categories come from small subcategories of Abelian categories such as categories of vector spaces, modules etc. and the concept of a normal, reductive category can be regarded as a generalization of the concept of semdsimplicity or rcducibility of.