Quasi-N4-Lattices

Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic , results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic ) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras . In the present paper we propose the class of quasi- N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

Adjoint Operations in Twist-Products of Lattices

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

Residuation in non-associative MV-algebras

It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

Separation axioms in \(L\)-fuzzifying supra-topology

In this paper, we define and investigate the notions of \(L\)-separation axioms in \(L\)-fuzzifying supra-topology. Also, some of their characterizations and a systematic discussion on the relationship among these notions is gave in \(L\)-fuzzifying supra-topology where \(L\) is a complete residuated lattice. Sometimes we need more conditions on \(L\) such as the completely distributive law or that the "\(\wedge\)" is distributive over arbitrary joins or the double negation law as we illustrate through this paper. As applications of our work the corresponding results (see \cite{2, 13}) are generalized and new consequences are obtained.

Residuation in orthomodular lattices

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.

When does a semiring become a residuated lattice?

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.