Homomorphisms and subdirect decompositions of semi-groups.
Homomorphisms and subdirect decompositions of semi-groups.
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DOI：
10.2140/pjm.1966.17.529
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发表时间：
1966-06
影响因子：
0.6
通讯作者：
B. Schein
中科院分区：
文献类型：
--
作者：
B. Schein
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 nonnull 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 (/^-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 /-regular rings [3] we introduce /-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 A, —>,, A 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 ε(g} is the ε-class containing g and Qi Ξ 02(ε) or 0i Ξ g2 means that gι and g2 are in the relation ε. If G is a semigroup then G denotes G with adjoined identity (unless G already has an identity), G° 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
