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Let P(X)P\left(X) be a partial transformation semigroup on a non-empty set XX. For a fixed non-empty subset YY of XX, let PT¯(X,Y)={α∈P(X)∣(domα∩Y)α⊆Y}.\overline{PT}\left(X,Y)=\left\{\alpha \in P\left(X)| \left({\rm{dom}}\hspace{0.33em}\alpha \cap Y)\alpha \subseteq Y\right\}. Then, PT¯(X,Y)\overline{PT}\left(X,Y) consists of all the mapping in P(X)P\left(X) that leave Y⊆XY\subseteq X as an invariant. It is a generalization of P(X)P\left(X) since PT¯(X,X)=P(X)\overline{PT}\left(X,X)=P\left(X). In this article, we present the necessary and sufficient conditions for elements of PT¯(X,Y)\overline{PT}\left(X,Y) to be regular, left regular, and right regular. The results are used to describe the relationships between these elements and determine their number when XX is a finite set. Moreover, we show that PT¯(X,Y)\overline{PT}\left(X,Y) is always abundant.

Let X n = { 1 , 2 , ... , n } and P n be a partial transformation semigroup on X n. Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \ { 0 }. An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α. The set Ann (α) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n. It is clear that 0 ∈ Ann (α) for all α ∈ P n. Let Z (P n) be the set of all zero divisors of P n and Z (P n) * = Z (P n) \ { 0 }. Generally, if α ∈ Z (P n) , then there exists β ∈ Z (P n) * in which β ∈ Ann (α). In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z (P n) * and two distinct vertices α and β are adjacent if and only if Ann (α) ∩ Ann (β) ≠ { 0 }. Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and T (X) denote the full transformation semigroup on X. For a fixed nonempty subset Y of X , let PG Y (X) = { α ∈ T (X) : α | Y ∈ G (Y) } , where G (Y) is a permutation group on Y. Then PG Y (X) is a regular submonoid of T (X). In this paper, we describe all intra-regular and unit regular elements of PG Y (X) and give necessary and sufficient conditions for PG Y (X) to be intra-regular and unit regular. We also count the number of these elements when X is a finite set. Moreover, we classify the maximal subsemigroups of PG Y (X) and prove that these maximal subsemigroup coincide with the maximal regular subsemigroups of PG Y (X) when X / Y is a finite set. [ABSTRACT FROM AUTHOR]

This paper explores the ideals and their structural properties in two generalizations of the partial transformation semigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated. [ABSTRACT FROM AUTHOR]

Given a non-empty set X and let P(X) be the partial transformation semigroup on X. For a fixed non-empty subset Y of X, let PFix(X, Y) = {α ∈ P(X) : yα = y for all y ∈ dom(α) ∩ Y }. Then PFix(X, Y) is a subsemigroup of P(X). In this paper, we show that PFix(X, Y) is always abundant, even though it is not regular. Moreover, unit regular and coregular elements of such semigroup are all completely characterized. [ABSTRACT FROM AUTHOR]

Let ΠL1 denote a direct power of L1, the two-element left zero semigroup with identity adjoined. A semigroup S is called left quasi-ample if for each a∈S there exists a unique idempotent a+∈S such that for all x, y∈S1 and the left ample condition holds. Generalizing a recent result in [3], we prove that the semigroups in the title are embeddable into certain transformation semigroups. Our embedding provides an easy way to construct (finite) proper covers for (finite) such semigroups. Moreover, we show that each proper such semigroup is embeddable into a semidirect product of a ΠL1-embeddable band by a right cancellative monoid, giving a partial answer to a question raised in [1]. [ABSTRACT FROM AUTHOR]