On the embedding preordering of universal algebras

A classical fact is that sets of subalgebras of a universal algebra ordered by inclusion are, up to order isomorphism, the complete compactly generated lattices. We investigate the sets of subalgebras of a universal algebra, and moreover, various classes of algebras, preordered by embeddings. Such preorders have a much more complicated structure. Though we do not have their complete characterization yet, the results that we have obtained show that there are three essentially distinct cases depending on whether the algebra under consideration has a single unary operation, or only unary operations and at least two of those, or at least one operation of larger arity. We apply our results to calculate the modal logics associated with the preordered structures of subalgebras of an algebra. In these logics, the possibility of a formula means that the formula holds in some subalgebra.