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Title page for ETD etd-03312003-141423
|Type of Document
|Author's Email Address
||Varieties of residuated lattices
- residuated lattices
- subvariety lattice
|Date of Defense
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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