Varieties with cofinal sets: examples and amalgamation

Abstract

A variety $\mathcal {V}$ has a cofinal set $S \subset \mathcal {V}$ if any $A \in \mathcal {V}$ is embeddable in a reduced product of members of $S$. Amalgamation in and examples of such varieties are considered. Among other results, the following are proved: (i) every lattice is embeddable in an ultraproduct of finite partition lattices; (ii) if $\mathcal {V}$ is a residually small, congruence distributive variety whose members all have one-element subalgebras, then the amalgamation class of $\mathcal {V}$ is closed under finite products.