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Title page for ETD etd-03312003-141423


Type of Document Dissertation
Author Galatos, Nikolaos
Author's Email Address ngalatos@math.vanderbilt.edu
URN etd-03312003-141423
Title Varieties of residuated lattices
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Constantine Tsinakis Committee Chair
Alan Peters Committee Member
Jonathan Farley Committee Member
Ralph McKenzie Committee Member
Steven Tschantz Committee Member
Keywords
  • variety
  • distributive
  • residuated lattices
  • cancellative
  • MV-algebras
  • subvariety lattice
Date of Defense 2003-04-17
Availability unrestricted
Abstract
A residuated lattice is an algebraic structure that has a lattice and a monoid reduct, such that multiplication is residuated with respect to the order. Residuated lattices generalize many well studied algebras including lattice-ordered groups, Brouwerian algebras and generalized MV-algebras. Moreover, they are connected to sub-structural logic, since they constitute algebraic semantics for the unbounded full Lambek calculus.

Residuated lattices form a variety. We investigate the lattice of its subvarieties and concentrate on a number of interesting subvarieties. In particular, we construct a continuum of atomic varieties and prove that the join of two finitely based commutative residuated-lattice varieties is also finitely based. Moreover, we study the varieties of cancellative and of distributive residuated lattices and present a duality theory for the bounded members of the latter. Finally, we generalize standard MV-algebras and describe a representation theorem and a categorical equivalence about them. As a corollary we obtain the decidablility of their equational theory.

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