*Article* **Left (Right) Regular Elements of Some Transformation Semigroups**

**Kitsanachai Sripon 1, Ekkachai Laysirikul 1,\* and Worachead Sommanee <sup>2</sup>**


**Abstract:** For a nonempty set *X*, let *T*(*X*) be the total transformation semigroup on *X*. In this paper, we consider the subsemigroups of *T*(*X*) which are defined by *T*(*X*,*Y*) = {*α* ∈ *T*(*X*) : *Xα* ⊆ *Y*} and *S*(*X*,*Y*) = {*α* ∈ *T*(*X*) : *Yα* ⊆ *Y*} where *Y* is a non-empty subset of *X*. We characterize the left regular and right regular elements of both *T*(*X*,*Y*) and *S*(*X*,*Y*). Moreover, necessary and sufficient conditions for *T*(*X*,*Y*) and *S*(*X*,*Y*) to be left regular and right regular are given. These results are then applied to determine the numbers of left and right regular elements in *T*(*X*,*Y*) for a finite set *X*.

**Keywords:** regular elements; magnifying elements; transformation semigroups

**MSC:** 20M20
