Identical inclusions of semilattices (Q2182565)

Let \({X^ + }\) be the free semigroup generated by a set \(X\), \(u \in {X^ + }\), \(V \subseteq {X^ + }\). A semigroup \(S\) satisfies an identical inclusion \(u\hat \in V\) iff \((\exists v \in V)(\varphi (u) = \varphi (v))\) for every mapping \(\varphi :X \to S\). An identical inclusion is called elementary if the set \(V\) is finite. Here, the author shows that any identical inclusion of a semilattice is elementary and obtains a complete syntactic, semantic and algorithmic characterization of inclusive varieties of semilattices.