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Title page for ETD etd-05032013-145420


Type of Document Dissertation
Author Wires, Alexander Duane
Author's Email Address slawkenbergius@hotmail.com
URN etd-05032013-145420
Title Some Results in Universal Algebra
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Ralph McKenzie Committee Chair
Constantine Tsinakis Committee Member
Mark Ellingham Committee Member
Steven Tschantz Committee Member
Yaqiong Xu Committee Member
Keywords
  • Definability
  • substructure ordering
  • simple graphs
  • weak difference term
  • equivalence relations
Date of Defense 2013-04-23
Availability unrestricted
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.

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