Some Results in Universal Algebra
Wires, Alexander Duane
:
2013-05-28
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.