Normal band compositions of semigroups
Miroslav Ćirić, Stojan Bogdanović · Proceedings of the Japan Academy Series A Mathematical Sciences · 1993
In this paper we give a construction of bands of arbitrary semigroups and we apply this result to study of normal bands of semi- groups, and especially for normal bands of monoids.We generalize some well-known results concerning normal bands of monoids and groups.In this paper we consider band compositions in the general case.Using a general construction for a semilattice of semigroups, we give a construction for a band of arbitrary semigroups.This construction is a very simple con- sequence of Theorem A, but we give some important applications of this con- struction: We give a description of normal bands of arbitrary semigroups, especially of normal bands of monoids, and as consequences we obtain some well-known results concerning normal bands of monoids and groups.Note that in our considerations, the conditions ( 5) and ( 6) in Theorem A have the important role.Throughout this paper, S (B;Si) means that a semigroup S is a band B of semigroups Si, B. Let S (B;S), where each S is a monoid with the identity e, S is a systematic band B of S, i B, if ij j ee e and ji j=:> ee e (M.Yamada [141).S is a proper band of Si if (eli B} is a subsemigroup of S (B.M. Schein [11]).Let S be an ideal of a Semigroup D. A congruence a on D is an S-congruence on D if its restrictidn on S is the equality relation on S.An ideal extension D of a semigroup S is a dense extension of S if the equality relation is the unique S-congruence on D.Theorem A [9].Let Y be a semilattice.For each o Y we associate a semigroup S and an extension D of S such that D f? D 0 if c :/: .F or every pair a, fl Y such that c >_ fl let qba., Sa D, be a mapping satisfying:(1) Ca.a is the identity mapping on Sa(2) (S,) (S,) _ S (3) [(a,a) (ba,.z)].e,(a.,r) (bCa,r), for all c, fl, )" Y such that aft > )" and all a S a, b Sa.Define a multiplication on S U arSa with:(4) a* b (a,.)(bCa,a), (a S., b Sa).Then S is a semilattice Y of semigroups Sa, in notation S (Y S a, Ca,, Da).Conversely. every semigroup S which is a semilattice Y of semigroups S a can be so constructed.In addition.D a can be chosen to satisfy:(5) D {ba,.fl >-, b Sa, fl Y} (6) D. is a dense extension of S a. *)