Introduction to semigroup theory

---------------------------------------------------------------------***--------------------------------------------------------------------Abstract A fuzzy subset μ of a group G is called a fuzzy subgroup of G if (i) the fuzzy value μ(xy) of the product of any two elements x,y of G is not less than the minimum of the fuzzy values μ(x), μ(y), and (ii) the fuzzy value μ(x-1) of the inverse of an element x is equal to the fuzzy value μ(x) of the element. We prove that second condition can be derived from the first if and only if the group is torsion

On the definition of fuzzy subgroup

A new type of fuzzy semiprime subsets in ordered semigroups

We establish a necessary condition ( E ) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition ( S ) for a variety to consist entirely of saturated semigroups. Condition ( S ) is shown to be sufficient for heterotypical varieties and a stronger condition ( S ′) is shown to be sufficient for homotypical varieties.

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Saturated and epimorphically closed varieties of semigroups

We determine which permutative varieties are saturated and classify all nontrivial permutation identities for the class of all globally idempotent semigroups.

/pdf/on-saturated-permutative-varieties-and-consequences-of-3xegy8dltx.pdf

On saturated permutative varieties and consequences of permutation identities

Epimorphisms, dominions and varieties of semigroups

Ordered semigroups characterized in terms of generalized fuzzy ideals

/pdf/epimorphically-closed-permutative-varieties-42j51tczw8.pdf

Epimorphically closed permutative varieties

In this paper, some properties of the ( k ∗ , k ) -lower part of ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals are obtained. Then, we characterize regular ordered semigroups in terms of its ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy generalized bi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left ideals and ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy right ideals, and an equivalent condition for ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left (resp. right) ideals is obtained. Finally, the existence theorems for an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal as well as for the minimality of an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal of an ordered semigroup are provided.

https://www.mdpi.com/2227-7390/7/5/401/pdf

Noor Mohammad Khan

Papers

A new type of fuzzy semiprime subsets in ordered semigroups

Epimorphisms, dominions and varieties of semigroups

Ordered semigroups characterized in terms of generalized fuzzy ideals

Epimorphically closed permutative varieties

Characterizations of Regular Ordered Semigroups by (∈,∈∨(k∗,qk))-Fuzzy Quasi-Ideals