Closure (interior) operators and closure systems are important tools in many mathematical environments. Considering the logical sense of a complete residuated lattice L, this paper aims to present concepts L-closure (L-interior) by means infimums (supremums) L-families L-subsets show their equivalence categorical sense. Also, two types fuzzy relations between corresponding L-interior proposed, ...

The theory of residuated lattices, first proposed by Ward and Dilworth [4], is formalised in Isabelle/HOL. This includes concepts of residuated functions; their adjoints and conjugates. It also contains necessary and sufficient conditions for the existence of these operations in an arbitrary lattice. The mathematical components for residuated lattices are linked to the AFP entry for relation al...

We extend the lattice embedding of the axiomatic extensions of the positive fragment of intuitionistic logic into the axiomatic extensions of intuitionistic logic to the setting of substructural logics. Our approach is algebraic and uses residuated lattices, the algebraic models for substructural logics. We generalize the notion of the ordinal sum of two residuated lattices and use it to obtain...

My primary and current research work is in the general spirit of algebras of logic, lattice theory, hyper structures and applications. A partially ordered set, or poset for short is a pair (P, ≤) where P is a set and ≤ a partial order on P. A poset (P, ≤) is called a lattice if every pair x, y ∈ P has a least upper bound x ∨ y and a greatest lowest bound x ∧ y in P. A lattice is bounded if it h...

It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to th...

We introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize ...

In the current study, we define different L-valued operators on LB-valued general fuzzy automata or simply GFA, where B is considered as a complete infinitely distributive lattice of propositions about GFA. particular, this study demonstrates that successor and predecessor induce co-topologies while residuated approximation topologies state set given Further, show continuity separation properti...

The so called Fuzzy Logic in Narrow Sense (FLn) is the basic constituent of mathematical fuzzy logic. Its development has been initiated by the paper of J. A Goguen [7] and the first, highly sophisticated paper is that of J. Pavelka [24]. His work has been continued by the author of this paper in [14] and especially in the book [13]. Fuzzy logic has been established as a generalization of class...

In this paper, we introduce a new algebra called ‘EQ-algebra’, which is an alternative algebra of truth values for formal fuzzy logics. It is specified by replacing implication as the main operation with a fuzzy equality. Namely, EQ-algebra is a semilattice endowed with a binary operation of fuzzy equality and a binary operation of multiplication. Implication is derived from the fuzzy equality ...

Hájek in [5] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structure of fuzzy contexts, see [2,6,9,10]. Recently, Molodtsov in [13] introduced the soft set as a mathematical tool for dealing information as the uncertainty of data in engineering, physics, computer sciences and many other diverse ...