Semigroups of partial transformations with kernel and image restricted by an equivalence

Abstract

For an arbitrary set X and an equivalence relation µ on X, denote by P µ (X) the semigroup of partial transformations α on X such that xµ ⊆ x(ker(α)) for every x ∈ dom(α), and the image of α is a partial transversal of µ. Every transversal K of µ defines a subgroup G = G K of P µ (X).We study subsemigroups G, U of P µ (X) generated by G ∪ U , where U is any set of elements of P µ (X) of rank less than |X/µ|. We show that each G, U is a regular semigroup, describe Green's relations and ideals in G, U , and determine when G, U is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top J -class J of P µ (X) is a right group. We find formulas for the ranks of the semigroups J, G ∪ I, J ∪ I, and I, where I is any proper ideal of P µ (X).

On Presentations of Semigroup of Transformations Restricted by an Equivalence

“…Yan and Wang [9] characterized the Greens relations (with respect to a nonempty subset U of the set of idempotents) as a new method of partition due to Lawson [10] on E(X, σ) where they proved among other results that the semigroup E(X, σ) is right Ehreshmann and the regular part, RE(X, σ) is orthodox (completely regular) if and only if the set X consists of at most two σ-classes. Andre and Konieczny (2020), [11] considered some generalizations of E(X, σ) as they studied the semigroup of partial transformations with restricted kernel and image.…”