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...

In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators.

In this paper, we investigate the properties of antitone Galois connection and formal concepts. Moreover, we show that order reverse generating maps induce formal, attribute oriented and object oriented concepts on a complete residuated lattice.

A b s t r a c t. In this paper, we prove the semisimplicity of free biresiduated lattices, more precisely, integral residuated lattices. In [4], authors show that variety of residuated lattices, more precisely, commutative integral residuated lattices, is generated by its finite simple members. The result is obtained by showing that every free residuated lattice is semisimple and then showing t...

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct. A residuated lattice is an algebra A = (A,∨,∧, ·, e, /, \) such that (A,∨,∧) is a lattice, (A, ·, e) is a monoid and for every a, b, c ∈ A ab ≤ c ⇔ a ≤ c/b ⇔ b ≤ a\c. The last condition is equivalent to the fact that (A,∨,∧, ·, e) is a lattice-ordered monoid and for every a, b ∈ A there is a great...

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the so-called pseudo-prelinear tri...

Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, followed by some results and open problems about these partially ordered algebras. Next we show that all complete homomorphic images of a complete residuated lattice A can be constructed easily on certain definable subsets of A. Applying these observations to the algebras of Hajek’s Bas...

In this paper, for a complete residuated lattice L, we present the categorical properties of ?-neighborhood spaces and their relationships to neighborhood stratified L-neighborhood spaces. The main results are: (1) category is topological category; (2) can be embedded in as reflective subcategory, when L meet-continuous lattice, subcategory; (3) continuous (resp., spaces) simultaneously corefle...

In this paper, the concept of intuitionistic fuzzy congruence relation on a residuated lattice is introduced and its properties is studied. The relationship between intuitionistic fuzzy filters and intuitionistic fuzzy congruence relations on a residuated lattice is obtain. Then the intuitionistic fuzzy congruence relation corresponding to a given intuitionistic fuzzy filter on residuated latti...

Morphisms of some categories of sets with similarity relations (Ω-sets) are investigated, where Ω is a complete residuated lattice. Namely a category SetF(Ω) with morphisms (A, δ) → (B, γ) defined as special maps A → B and a category SetR(Ω) with morphisms defined as a special relations A × B → Ω. It is proved that arbitrary maps A → Ω and A × B → Ω can be extended onto morphisms (A, δ) → (Ω,↔)...