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 AbstractA 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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