The class of prime semilattices is not finitely axiomatizable (Q1362896)

A meet-semilattice \(S\) is prime iff for \(x\nleq y\) in \(S\) there is a prime filter in \(S\) containing \(x\) and omitting \(y\). \textit{R. Balbes} [Trans. Am. Math. Soc. 136, 261-267 (1969; Zbl 0175.01402)] has shown that \(S\) is prime iff binary meet distributes over all finite existing joins. \textit{B. M. Schein} [Algebra Univers. 2, 1-2 (1972; Zbl 0248.06006)] stated without proof that the class of prime semilattices is not finitely axiomatizable. The purpose of this note is to validate Schein's claim. To this end, the author constructs a sequence of first-order sentences \(D_n\) expressing that binary meet distributes over existing \(n\)-ary joins and then shows that \(D_n\) does not imply \(D_{n+1}\) for any \(n\geq 1\). It follows that no finite collection of \(D_n\)'s will axiomatize prime semilattices and this class, consequently, cannot be finitely axiomatizable.