Vol. 17, No. 3, 1966

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Homomorphisms and subdirect decompositions of semi-groups

Boris M. Schein

Vol. 17 (1966), No. 3, 529–547

Subdirect decompositions of rings seem to be an important tool in the theory of rings promoting the development of this theory. It is a very natural thing to study subdirect products of semigroups but to the author’s knowledge the only paper on the topic is that of G. Thierrin [22] where certain properties of subdirectly irreducible semigroups are considered.

Subdirect decompositions of semigroups are closely connected with homomorphisms of these semigroups, so we describe in the first section the structure of an arbitrary congruence on a semigroup. The second section is devoted to certain special subsets and elements of a semigroup. Main notions of the section are those of disjunctive element (i.e., an element that does not form a congruence class modulo any nontrivial congruence) and of core of a semigroup (i.e., a least nonnulI ideal). Subdirectly irreducible semigroups are considered in the third, fourth and fifth sections. We consider certain general properties of such semigroups and find characterizations of special classes of such semigroups (e.g. nilpotent, idempotent, commutative). Section 6 treats homomorphically simple (h-simple) semigroups, i.e., semigroups having no nontrivial congruences. Section 7 is devoted to consideration of certain semigroups having special subdirect decompositions. By analogy with f-regular rings [3] we introduce f-regular semigroups. There are considered also completely reductive semigroups, i.e., semigroups having no nononreductive homomorphic images.

Several results of this paper have been published without proofs in our note [18]. Certain results of [18] had been previously found in [22] but we did not know this when [18] was published. All concepts of the theory of semigroups that are not defined here are defined in [6, 12]. We use the symbols Λ,,, respectively for conjunction, implication, (logical) equivalence, universal quantifier and follow the ordinary agreement as to the use of brackets in statements. If 𝜖 is an equivalence relation, then 𝜖gis the 𝜖-class containing g and g1 g2(𝜖) or g1 g2 means that g1 and g2 are in the relation 𝜖. If G is a semigroup lhen G1 denotes G with adjoined identity (unless G already has an identity), GO denotes G with adjoined zero (unless G already has a zero). Variables g and h (with or without indices) take values in the set of all elements of G, variables x and y take values in the set of all elements of G1 (i.e., x and y may be void symbols ([12], p. 7)). A one-element set is often denoted in the same way as its element. As a rule, one-element semi-groups are excluded from consideration. ΔG is the identity relation on the set G.

Mathematical Subject Classification
Primary: 20.92
Received: 9 October 1964
Published: 1 June 1966
Boris M. Schein