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Remarks on ideals in lower-bounded dually residuated lattice-ordered monoids

Jan Kühr (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Lattice-ordered groups, as well as G M V -algebras (pseudo M V -algebras), are both particular cases of dually residuated lattice-ordered monoids ( D R ℓ -monoids for short). In the paper we study ideals of lower-bounded D R ℓ -monoids including G M V -algebras. Especially, we deal with the connections between ideals of a D R ℓ -monoid A and ideals of the lattice reduct of A .

Representable dually residuated lattice-ordered monoids

Jan Kühr (2003)

Discussiones Mathematicae - General Algebra and Applications

Dually residuated lattice-ordered monoids (DRl-monoids) generalize lattice-ordered groups and include also some algebras related to fuzzy logic (e.g. GMV-algebras and pseudo BL-algebras). In the paper, we give some necessary and sufficient conditions for a DRl-monoid to be representable (i.e. a subdirect product of totally ordered DRl-monoids) and we prove that the class of representable DRl-monoids is a variety.

Representation and duality for Hilbert algebras

Sergio Celani, Leonardo Cabrer, Daniela Montangie (2009)

Open Mathematics

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are...

Residual implications and co-implications from idempotent uninorms

Daniel Ruiz, Joan Torrens (2004)

Kybernetika

This paper is devoted to the study of implication (and co-implication) functions defined from idempotent uninorms. The expression of these implications, a list of their properties, as well as some particular cases are studied. It is also characterized when these implications satisfy some additional properties specially interesting in the framework of implication functions, like contrapositive symmetry and the exchange principle.

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