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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 𝜖⟨g⟩ is 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.
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