We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison with similar structures such as residuated lattices and concurrent Kleene algebras. Introducing the notion of an ideal with respect to the compositional prod...

Sugeno integral is one of the basic aggregation operations on a qualitative scale where only minimum and maximum, as well as order-reversing maps are allowed. Recently some variants of this aggregation operation, named soft and drastic integrals, have been introduced in a previous work by three of the authors. In these operations, importance weights play the role of tolerance thresholds enablin...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

To express wider uncertainty, B?hounek and Da?ková studied fuzzy partial logic function. At the same time, Borzooei generalized t-norms put forward concept of when studying lattice valued quantum effect algebras. Based on t-norms, Zhang et al. residuated implications (PRIs) proposed lattices (PRLs). In this paper, we mainly study related algebraic structure logic. First, provide definitions reg...

We show that every poset P with an antitone involution can be extended to a commutative integral residuated E(P). If, moreover, is lattice then so

Gentzen systems are introduced for Spinks and Veroff’s substructural logic corresponding to constructive logic with strong negation, and some logics in its vicinity. It has been shown by Spinks and Veroff in [9], [10] that the variety of Nelson algebras, the algebras of constructive logic with strong negation N, is term-equivalent to a certain variety of bounded commutative residuated lattices ...

In this paper, we introduce the notion of co-annihilator of a subsetin a triangle algebra. It is shown that the co-annihilator of asubset is an interval valued residuated lattice (IVRL)-filter. Also, aspecial set of a triangle algebra is defined and the relationshipbetween this set and co-annihilator of a subset in triangle algebrais considered. Finally, co-annihilators preserving congruencerel...