| 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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| Filename |
Size |
Approximate Download Time
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| |
NGdissertation.pdf |
574.98 Kb |
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