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Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

100%

Dzik W. , Radeleczki S.

Bulletin of the Section of Logic

2016

tom 45

nr 3/4

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

Quantitative Analysis of Lattice-valued Kripke Structures

100%

Pan H. , Zhang M. , Wu H. , Chen Y.

Fundamenta Informaticae

2014

tom Vol. 135, nr 3

269--293

To model and analyze systems with multi-valued information, in this paper, we present an extension of Kripke structures in the framework of complete residuted lattices, which we will refer to as lattice-valued Kripke structures (LKSs). We then show how the traditional trace containment and equivalence relations, can be lifted to the lattice-valued setting, and we introduce two families of lattice-valued versions of the relations. Further, we explore some interesting properties of these relations. Finally, we provide logical characterizations of our relations by a natural extension of linear temporal logic.

Interior and closure operators on bounded commutative residuated l-monoids

100%

Rachůnek J. , Švrček F.

Discussiones Mathematicae - General Algebra and Applications

2008

tom 28

nr 1

11-27

Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the Hájek's basic fuzzy logic) and Heyting algebras as proper subclasses.

Residuation in orthomodular lattices

86%

Chajda I. , Länger H.

Topological Algebra and its Applications

2017

tom 5

nr 1

1-5

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.

The reticulation of residuated lattices induced by fuzzy prime spectrum

86%

Ghorbani S.

2012

tom Vol. 35

23--32

In this paper, we use the fuzzy prime spectrum to define the reticulation (L(A),λ) of a residuated lattice A. We obtain some related results. In particular, we show that the lattices of fuzzy filters of a residuated lattice A and L(A) are isomorphic and the fuzzy prime spectrum of A and L(A) are homomorphic topological space.