Residuated lattice orders on cancellative monoids

A residuated lattice is an algebra that combines a monoid structure and a lattice structure through the use of left and right residuals. This important class of algebras contains both lattice-ordered groups and Boolean algebras. They have also been studied in the context of logic, image analysis, and formal languages.; The main results in this thesis are concerned with cancellative residuated lattices. We first extend the work of Saito, et al. on cancellative total orders to the case of cancellative residuated chains. This gives us a way to describe any residuated total order on a finitely generated monoid. We are also able to use this work to find a necessary condition to construct any residuated total order on a finitely-generated commutative cancellative free monoid. This condition is also sufficient when the monoid is a free commutative monoid, and we are able to construct any residuated total order on such a monoid. We end this section with a discussion on ordering the direct product of two such residuated chains.; We also investigate the lattice reducts of cancellative residuated lattices. We show that any lattice can be embedded into a cancellative residuated lattice. This construction also gives an order embedding from the variety of all lattices into the variety of cancellative residuated lattices. We also study whether these results can be extended to the commutative case, and we are able to show that several lattices can be embedded into such a residuated lattice.