Type of Document Dissertation Author Wires, Alexander Duane Author's Email Address email@example.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
- substructure ordering
- simple graphs
- weak difference term
- equivalence relations
Date of Defense 2013-04-23 Availability unrestricted AbstractIn 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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