Abstract

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

Highlights

  • The graph of an algebra A = (A, O) is the relational structure G(A) = A, {Ro | o ∈ O}, Communicated by Mikhail Volkov

  • In [15, Theorem 1] it is proved that there is no finite basis for the quasi-equational theory of G(C ) whenever C is a class of semigroups possessing a nontrivial member with a neutral element, that is, an element e such that ae = ea = a for all a. (The case when C consists of any individual twoelement semigroup with a neutral element was proved earlier by Gornostaev in [8], see [7, Sect. 6.2].) Here we indicate that this shortcoming of relational structures, compared to algebras, carries over to the topological setting

  • Following Clark, Davey, Jackson and Pitkethly [6], we define the Boolean core HBC of a universal Horn class H as the topological prevariety generated by the finite members of H

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Summary

Introduction

Following Clark, Davey, Jackson and Pitkethly [6], we define the Boolean core HBC of a universal Horn class H as the topological prevariety generated by the finite members of H (treated as topological structures). We are interested in when HBC admits a first-order axiomatization (relative to all Boolean topological structures in the language of H ) With respect to this problem, our contribution is a proof of the following fact. Theorem 1.1 Let C be a class of semigroups possessing a nontrivial member with a neutral element and H be the universal Horn class generated by G(C ). One may show that the Boolean core of the class of graphs of semigroups satisfying (∀u, v)[u · v ≈ u] is axiomatizable by a finite number of universal Horn sentences

Semigroups
Relational structures
Background
Lack of standardness
Non-idempotent case
Semilattice case
Idempotent case
Graphs of monoids and groups
Full Text
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