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 the separtion axioms T0, T1, T2 and T5/2 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 α, there exists at least one nontrivial Hausdorff topological residuated lattice of cardinality α. In the follows, we obtain some conditions on (semi)...

we consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $x$, $(x, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $x$. we also show that, for a strong residuated lattice $x$, $bigcap {p_{lambda} ,|,p_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

the aim of this paper is to extend results established by h. onoand t. kowalski regarding directly indecomposable commutative residuatedlattices to the non-commutative case. the main theorem states that a residuatedlattice a is directly indecomposable if and only if its boolean center b(a)is {0, 1}. we also prove that any linearly ordered residuated lattice and anylocal residuated lattice are d...

in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete re...

Recently, Molodtsov [12] 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 field. Presently, the soft set theory is making progress rapidly [1,3]. Pawlak’s rough set [13,14] can be viewed as a special case of soft rough sets [3]. The topological structures of soft sets have been dev...

in this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. in particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. then we study fuzzy...

A notion of reticulation which provides topological properties on algebras has introduced on commutative rings in 1980 by Simmons in [5]. For a given commutative ring A, a pair (L, λ) of a bounded distributive lattice and a mapping λ : A → L satisfying some conditions is called a reticulation on A, and the map λ gives a homeomorphism between the topological space Spec(A) consisting of prime fil...

In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. Then we study fuzzy...