Semilattices of definable subalgebras. II

Abstract

In studying derived objects on universal algebras, such as automorphisms, endomorphisms, congruences, subalgebras, etc., we are naturally interested in those that can be defined by the means of the universal algebras themselves (i.e., are definable in one sense or another)—in particular, in which part of all relevant derived objects is constituted by these. It is proved that for any algebraic lattice L and any of its 0-1-lower subsemilattices L 0 ⊆ L 1 ⊆ L 2, there exist a universal algebra $ \mathcal{A} $ and an isomorphism φ of the lattice L onto the lattice Sub $ \mathcal{A} $ such that φ(L 0) = OFSub $ \mathcal{A} $ , φ(L 1) = POFSub $ \mathcal{A} $ , φ(L 2) = FSub $ \mathcal{A} $ , and PFSub $ \mathcal{A} $ = FSub $ \mathcal{A} $ .