We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

We prove that when divisibility is added to a residuated multilattice, this causes the multilattice structure to collapse down to a residuated lattice. This motivates the study of semi-divisibility and regularity on residuated multilattices. The ordinal sum construction is also applied to residuated multilattices as a way to construct new examples of both residuated multilattices and consistent...

‎We give a simple and independent axiomatization of reticulations on residuated lattices‎, ‎which were axiomatized by five conditions in [C‎. ‎Mureşan‎, ‎The reticulation of a residuated lattice‎, ‎Bull‎. ‎Math‎. ‎Soc‎. ‎Sci‎. ‎Math‎. ‎Roumanie‎ ‎51 (2008)‎, ‎no‎. ‎1‎, ‎47--65]‎. ‎Moreover‎, ‎we show that reticulations can be considered as lattice homomorphisms between residuated lattices and b...

In this paper, we study the separtion axioms $T_0,T_1,T_2$ and $T_{5/2}$ on topological and semitopological residuated lattices and we show that they are equivalent on topological residuated lattices. Then we prove that for every infinite cardinal number $alpha$, there exists at least one nontrivial Hausdorff topological residuated lattice of cardinality $alpha$. In the follows, we obtain some ...

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in re...

Ward and Dilworth 1 introduced the concept of residuated lattices as generalization of ideal lattices of rings. The residuated lattice plays the role of semantics for a multiple-valued logic called residuated logic. Residuated logic is a generalization of intuitionistic logic. Therefore it is weaker than classical logic. Important examples of residuated lattices related to logic are Boolean alg...

Proposition 3. If A=〈A,∧,∨, ·, \, /, 1, γ〉 is a residuated lattice with a nucleus γ, then the algebra Aγ=〈Aγ ,∧,∨γ , ·γ , \, /, γ(1)〉 is a residuated lattice, where Aγ=γ[A] and for all x, y ∈ Aγ , x ∨γ y=γ(x ∨ y) and x ·γ y=γ(x · y). For a variety V of residuated lattices with modal operators, consider V∗, the full subcategory of V consisting of those pairs 〈B, 〉 such that B generates B as a re...

In this paper, based on Hájek, Vychodil, Rachunek and Šalounová’s works, we study the concept of v-filters of residuated lattices with weak vt-operators, axiomatize very true operators, discuss filters and v-filters of residuated lattices with weak vt-operator, give the formulas for calculating the v-filters generated by subsets, and show that lattice of v-filters of a commutative residuated la...

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...