dc.creator | Wires, Alexander Duane | |
dc.date.accessioned | 2020-08-22T00:39:35Z | |
dc.date.available | 2015-05-28 | |
dc.date.issued | 2013-05-28 | |
dc.identifier.uri | https://etd.library.vanderbilt.edu/etd-05032013-145420 | |
dc.identifier.uri | http://hdl.handle.net/1803/12260 | |
dc.description.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. | |
dc.format.mimetype | application/pdf | |
dc.subject | Definability | |
dc.subject | substructure ordering | |
dc.subject | simple graphs | |
dc.subject | weak difference term | |
dc.subject | equivalence relations | |
dc.title | Some Results in Universal Algebra | |
dc.type | dissertation | |
dc.contributor.committeeMember | Steven Tschantz | |
dc.contributor.committeeMember | Constantine Tsinakis | |
dc.contributor.committeeMember | Mark Ellingham | |
dc.contributor.committeeMember | Yaqiong Xu | |
dc.type.material | text | |
thesis.degree.name | PHD | |
thesis.degree.level | dissertation | |
thesis.degree.discipline | Mathematics | |
thesis.degree.grantor | Vanderbilt University | |
local.embargo.terms | 2015-05-28 | |
local.embargo.lift | 2015-05-28 | |
dc.contributor.committeeChair | Ralph McKenzie | |