Some Results in Universal Algebra

Abstract

In the first part, we explore definability in the substructure relation. Let U denote either the universal class of irreflexive symmetric digraphs or equivalence relations. We analyze first-order definability in the ordered set of finite isomorphism types of structures in U ordered by embeddability. We prove the this ordered set has only one non-identity automorphism and each finite isomorphism type is definable up to to this automorphism. These results can be utilized to explore first-order definability in the closely associated lattice of universal subclasses of U . We show the lattice of universal subclasses has only one non-identity automorphism, the set of finitely generated and finitely axiomatizable universal subclasses are separately definable, and each such universal subclass is definable up to the unique non-identity automorphism; furthermore, we show that after adding a single constant type c, first-order definability in the substructure relation captures, up to isomorphism, second-order satisfiability among the finite structures in U . In the second part, we provide an alternate characterization for quasivarieties which extends the malcev condition for varieties with a weak difference term. As an application, we derive elementary proofs of two well-known results in the theory of digraph polymorphisms.