DEFINABILITY OF THE VARIETY GENERATED BY A COMMUTATIVE MONOID IN THE LATTICE OF COMMUTATIVE SEMIGROUP VARIETIES

Abstract. Let M be a commutative monoid. We construct a first-order formula that defines the variety generated by M in the lattice of all commutative semigroup varieties. A subset A of a lattice L; ∨, ∧ is called definable in L if there exists a firstorder formula Φ(x) with one free variable x in the language of lattice operations ∨ and ∧ which defines A in L. This means that, for an element a ∈ L, the sentence Φ(a) is true if and only if a ∈ A. If A consists of a single element, we speak about definability of this element. We denote the lattice of all commutative semigroup varieties by Com. A set of commutative semigroup varieties X (or a single commutative semigroup variety X ) is said to be definable if it is definable in Com. In this situation we will say that the corresponding first-order formula defines the set X or the variety X . Let M be a commutative monoid. In [10, Corollary 4.8], we provide an explicit first-order formula that defines the variety generated by M in the lattice of all semigroup varieties. The objective of this note is to modify the arguments from We will denote the conjunction by & rather than ∧ because the latter symbol stands for the meet in a lattice. Since the disjunction and the join in a lattice are denoted usually by the same symbol ∨, we use this symbol for the join and denote the disjunction by or. Evidently, the relations ≤, ≥, < and > in a lattice L can be expressed in terms of, say, meet operation ∧ in L. So, we will freely use these four relations in formulas. Let Φ(x) be a first-order formula