Varieties of commutative semigroups

Abstract

Abstract.In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.

Automorphisms of the lattice of equational theories of commutative semigroups

“…It is a nontrivial task to determine J knowing a base for a theory E. For an algorithm and examples the reader is referred to [8]. We shall not use it in this paper.…”

“…By results of [8] every theory generated by an irregular equation, as well as every theory generated by an equation (a, b) with comparable but nonequivalent sequences, is a Schwabauer theory. By Theorem 2.3 these theories are definable.…”

“…In contrast, we have very good knowledge about the structure of the lattice L(Com) of equational theories of commutative semigroups [8]. Nevertheless, this lattice seemed large and complicated enough so that one could believe that every individual theory is definable in it and that it has no nontrivial automorphisms.…”

“…So, it came as a surprise when Kisielewicz [9] (using the description in [8]) discovered that there exist nontrivial automorphisms of L(Com). A generalization of Kisielewicz's example by McKenzie has even shown that the group of automorphisms of L(Com) contains a Boolean group of order continuum.…”

“…It is proved in [8] that (Γ, ≤) is a well-quasi-ordered set, which means, in particular, that there are no infinite descending chains in Γ and every (order) filter J in Γ is finitely generated. A filter generated by the sequences a 1 , .…”