Structure of a Class of Regular Semigroups and Rings

One of the most natural approaches to the study of regular semigroups is to impose restrictions on the partial ordering of their idempotents (erg/^e/—jfe==e). The principal object of this note is to describe the structure of the classes of regular semigroups whose idempotents form a tree or a unitary tree, respectively (see Definition 1). We determine, among other things, a complete set of invariants, isomorphisms, the group of automorphisms, and congruences of these semigroups. We also consider regular rings whose multiplicative semigroup satisfies conditions (C) or (Ci) and give their structure. The terminology concerning semigroups is that of [2] and concerning p.o. sets of [ l ] . We consider only semigroups with zero; the statements concerning semigroups without zero can then be easily deduced. DEFINITION 1. A [unitary] tree T is a p.o. set with a unique minimal element 0 [and a unique maximal element l ] , r ^ { o | , satisfying (i) all elements [different from l ] are of finite height; (ii) every element different from 0 [and different from l ] covers exactly one element. DEFINITION 2. A regular semigroup whose p.o. set of idempotents is a tree [unitary tree] is said to be ^-regular [^regular]. A 3i-regular semigroup has an identity element. In order to find the structure of such semigroups, we need the following construction. For any semigroup S with zero, we write S* = S\0. Let T be a tree; to every aCzT*~T\Q associate a semigroup Sa with zero 0«; the semigroups Sa are pairwise disjoint. If ax>l (ax is the height of a in 7"), associate to a a partial homomorphism <*: S*—>S£ (â is the unique element of T covered by a). On the set V= (Uaesp* S*)\J0, multiplication is defined by induction on the height of aÇzT as follows. Let 0 act as the zero of V. If ax — fix — l and x£»S*, yÇzSp (multiplication in Sa is denoted by juxtaposition), let