Commutative semigroups with bounded orders of subdirectly irreducible acts

I B Kozhukhov

doi:10.22405/2226-8383-2021-22-1-188-199

Abstract

Subdirectly irreducible universal algebras, i.e., algebras which are not decomposable into non-trivial subdirect product, play an important role in mathematics due to well-known Birkhoff Theorem which states that any algebra is a subdirect product of subdirectly irreducible algebras (in another terminology: every algebra is approximated by the subdirectly irreducible algebras). In view of this, it seems reasonable to study algebras with certain conditions on subdirectly irreducible algebras. One of the natural restrictions is the finiteness of all subdirectly irreducible algebras. More stronger restriction is boundness of orders of all subdirectly irreducible algebras. An act over a semigroup (it is also an automaton, and a unary algebra) is a set with an action of the given semigroup on it. The acts over a fixed semigroup form a variety whose signature coincides with self semigroup. On the other hand, it is a category whose morphisms are homomorphisms from one act into another. It is not difficult to see that the semigroups over which all subdirectly ireducible acts are finite, are exactly the semigroups over which all subdirectly irreducible acts are finitely approximated (in another terminology: residually finite). A more narrow class form semigroups over which all acts are approximated by acts of n or less elements where n is a fixed natural number. In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice (a commutative idempotent semigroup). In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded. In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and also abelian groups over which all acts are approximated by acts of bounded orders. In this work, we characterize the commutative semigroups over which all acts are approximated by acts consisted of n or less elements.

Highlights

In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice
In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded
In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and abelian groups over which all acts are approximated by acts of bounded orders

Summary

Введение

Если полугруппа S имеет единицу (обозначим её через e) и xe = x для всех x ∈ X. Конгруэнция полигона X – это такое отношение эквивалентности ρ на множестве X, что (x, y) ∈ ρ → (xs, ys) ∈ ρ для всех x, y ∈ X, s ∈ S. Хорошо известно (и легко следует из определений), что нетривиальная алгебра A подпрямо неразложима в том и только том случае, если решётка ConA имеет наименьшую нетривиальную конгруэнцию. В работе [14] были описаны коммутативные подпрямо неразложимые полигоны (полигон X над полугруппой S называется коммутативным, если x(st) = x(ts) для всех x ∈ X, s, t ∈ S). В теореме 1 работы [16] доказывалось, что над полугруппой S все нетривиальные S-полигоны аппроксимируются двухэлементными в том и только том случае, если S – полурешётка Необходимые сведения из теории полугрупп можно найти в [1], теории полигонов в [15], универсальной алгебры в [8]

Предварительные результаты

Сведeние к точному полигону

NG-образ коммутативной периодической полугруппы

Окончание доказательства основной теоремы: достаточность