A lattice ordered monoid is a structure 〈L;⊕, 0L;≤〉 where 〈L;⊕, 0L〉 is a monoid, 〈L;≤〉 is a lattice and the binary operation ⊕ distributes over finite meets. If M ∈ R-Mod then the set ILM of all hereditary pretorsion classes of σ[M ] is a lattice ordered monoid with binary operation given by α :M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ ILM (the subscri...

Let M be a module over the commutative ring R. In this paper we introduce two new notions, namely strongly coprimal and super coprimal modules. Denote by ZR(M) the set of all zero-divisors of R on M . M is said to be strongly coprimal (resp. super coprimal) if for arbitrary a, b ∈ ZR(M) (resp. every finite subset F of ZR(M)) the annihilator of {a, b} (resp. F ) in M is non-zero. In this paper w...

Following the idea of $L$-fuzzy neighborhood system as introduced byFu-Gui Shi, and its generalization to $(L,M)$-fuzzy neighborhood system, the relationship between  $(L,M)$-fuzzy topology and $(L,M)$-fuzzy neighborhood system will be further studied. As an application of the obtained results, we will describe the initial structures of $(L,M)$-fuzzy neighborhood subspaces and $(L,M)$-fuzzy top...

Moreover the term predicate fuzzy logic L∀ will denote the predicate calculus obtained from the propositional core fuzzy logic L by adding the “classical” quantifiers (∀ and ∃) together with the same axioms and rule of generalization used by Hàjek in his book [7] when defining BL∀. On the other hand, given a left-continuous t-norm ∗, if L∗ denotes the (propositional) fuzzy logic which is standa...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

in this paper, we introduce some properties of an intuitionistic normal fuzzy m-subgroup of m- group with m-homomorphism and isomorphism. we study he image, the pre-image and the inverse mapping of the intuitionistic normal fuzzy m-‎subgroups.‎

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we...

Let D and G be copies of the open unit disc in C, let A (resp. B) be a measurable subset of ∂D (resp. ∂G), let W be the 2-fold cross ( (D ∪A)×B ) ∪ ( A× (B ∪G) ) , and let M be a relatively closed subset of W. Suppose in addition that A and B are of positive one-dimensional Lebesgue measure and that M is fiberwise polar (resp. fiberwise discrete) and that M ∩(A×B) = ∅. We determine the “envelop...

Milk thistle (Silybum marianum) is a native plant of Mediterranean Sea areas. Silymarin, an extract from its seeds, is used in various liver diseases for its hepatoprotective effects. It contains silymarin flavonoids (SF) such as silybin, silydianin and silychristin. Cyclosporin A (CsA) is widely used as an immunosuppressant in organ transplantations. Its other applications in many autoimmune d...

Let D=(V,A) be a digraphs without isolated vertices. The first Zagreb index of digraph D is defined as summation over all arcs, M1(D)=12∑uv∈A(du++dv−), where du+(resp. du−) denotes the out-degree (resp. in-degree) vertex u. In this paper, we give maximal values and set orientations unicyclic graphs with n vertices matching number m (2≤m≤⌊n2⌋).