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Title page for ETD etd-03132014-114415


Type of Document Dissertation
Author Smedberg, Matthew Raine
URN etd-03132014-114415
Title Necessary conditions for finite decidability in locally finite varieties admitting strongly abelian behavior
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Ralph McKenzie Committee Chair
Constantine Tsinakis Committee Member
Denis Osin Committee Member
Mark Sapir Committee Member
Miklos Maroti Committee Member
Keywords
  • strongly abelian
  • finite decidability
  • computability
  • locally finite varieties
Date of Defense 2014-03-11
Availability unrestricted
Abstract
We show that several kinds of local behavior in a finite algebra A present obstructions to the decidability of the first-order theory of the finite members of HSP(A). In particular, we show that every solvable congruence in a locally finite, finitely decidable variety is abelian, and that the subdirectly irreducible algebras in such a variety have very constrained congruence geometry, generalizing results of Idziak, Valeriote, and Willard for congruence-modular varieties. We then show that every finitely generated, finitely decidable variety is residually finite (indeed, has a finite residual bound). Finally we modify a construction of Valeriote to give a tighter bound on the essential arity of a sigma-sorted term operation of A.
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