Automorphisms of the lattice of equational theories of commutative semigroups

Abstract

Abstract. In this paper we complete the study of the first-order definability in the lattice of equational theories of commutative semigroups started by A. Kisielewicz in [Trans. Amer. Math. Soc. 356 (2004), 3483-3504]. We describe the group of automorphisms of this lattice and characterize firstorder definable theories, thus solving the problems posed by R. McKenzie and A. Kisielewicz.

The structure and definability in the lattice of equational theories of strongly permutative semigroups

“…In [6], we have completed the study of the first-order definability in the lattice L(Com). We have obtained complete descriptions of the set of definable theories and of the group of automorphisms of L(Com), which has appeared to form an uncountable Boolean group.…”

Definability for equational theories of commutative groupoids

“…Clearly, G a is a subgroup of the symmetric group on S(a). (See [2] for an exact description of G a ).…”

On definability in some lattices of semigroup varieties

“…However, he discovered that there exist nontrivial automorphisms of COM. In [2], the first-named author of the present paper completed the study of first-order definability in COM. He obtained complete descriptions of the set of definable varieties and of the group of automorphisms of COM, which turned out to be an uncountable Boolean group.…”

“…10, we consider varieties of the form var{u ≈ v, B ,r,n } where u ≈ v is a regular identity such that the words u and v are either equivalent or incomparable in the order ≤ B ,r,n and their lengths are shorter than n + + r. By translating and generalizing methods used in [2,3,12] we prove (Theorems 10.1 and 10.2) that each variety of this form is semi-definable in L(var B ,r,n ) if = r and definable in L(var B ,r,n ) if = r.…”

“…Acknowledgements O. Sapir thanks Dr. Ralph McKenzie for directing her attention to the articles [2,3] and Dr. Boris Vernikov for teaching her this subject and for many illuminating discussions. The authors are also grateful to Vyacheslav Shaprynskii for finding a hole in the previous version.…”