On smallest (generalized) ideals and semilattices of (2,2)-regular non-associative ordered semigroups

An ordered AG-groupoid can be referred to as a non-associativeordered semigroup, as the main di¤erence between an ordered semigroup and anordered AG-groupoid is the switching of an associative law. In this paper, wede ne the smallest left (right) ideals in an ordered AG-groupoid and use them tocharacterize a (2; 2)-regular class of a unitary ordered AG-groupoid along with itssemilattices and (2 ;2 _q)-fuzzy left (right) ideals. We also give the conceptof an ordered A*G**-groupoid and investigate its structural properties by usingthe generated ideals and (2 ;2 _q)-fuzzy left (right) ideals. These concepts willverify the existing characterizations and will help in achieving more generalizedresults in future works.

New types of fuzzy (m, n)-ideals in ordered semigroups

In this paper, we introduce new types of fuzzy (m, n)-ideals in ordered semigroups. In fact, the notion of (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals of the ordered semigroups is introduced. Further, we present the characterzations of this notion in different ways. Then the (κ*, κ)-lower part of the (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals is defined and its associated properties are investigated. After that, (m, n)-regular ordered semigroups are characterized in terms of its (∈ , ∈ ∨ (κ*, qκ))-fuzzy (m, n)-ideals and their (κ*, κ)-lower parts.

Sums and products of intervals in ordered semigroups

We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b. The multiplicative version of the above example is shown too. The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g ( x ) : = max { y ∈ ℤ + | I y ⊆ I x ⋅ I x } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev. Finally, we formulate some questions concerning the above topics.

A Benchmark Generalization of Fuzzy Soft Ideals in Ordered Semigroups

In real life, variability and inaccuracy are always presentand must be calculated by either possibilistic, probabilistic, polymorphic or other uncertainty approach. This benchmark study is about to construct new types of fuzzy soft ideals i.e., (∈, ∈ ∨qk )-FSR(L)Is of ordered semigroup(OSG). Based on this inception, fuzzy soft level subsets are defined which link ordinary ideals with (∈, ∈ ∨qk )-fuzzy soft left(right) ideals. Some binary operations like ◦ λ , intersection ∩ λ and union of fuzzy soft sets ∪ λ are given and various fundamental results of ideal theory are developed through these types of fuzzy soft ideals.

Some New Concepts on int-soft Ideals in Ordered Semigroups

In the present paper, we introduce some new notions on ordered semigroup. In fact, notion of a convex soft set in an ordered semigroup is introduced, and its basic properties are investigated. Moreover, we consider a characterization of a convex soft set. Furthermore, relations between a convex soft set and an int-soft [Formula: see text]-ideal (or, int-soft [Formula: see text]-ideal) are studied. Finally, int-soft [Formula: see text]-ideals (or, int-soft [Formula: see text]-ideals) generated by an ordered soft point are established.