อันดับธรรมชาติบนกึ่งกรุปของการแปลงที่คงสภาพEในตัว

Abstract

The objectives of this paper, we study a necessary and sufficient condition of natural order on a semigroup TsE ( x ). We also find maximal and minimal elements of TsE ( x ) .Let T( x ) be the full transformation semigroup on a set X ≠ Ø and E be an equivalence relation on X . In [1] show that the set TsE ( x ) ={f ∈ T( x ) | (x,xf)∈ E, ∀x ∈X } is a regular subsemigroup of T( x ) . In [2], the natural order on a regular semigroup is defined by a≤b if and only if eb = a = bf for some e ,f ∈ E(S ), where E(S) is the set of all the idempotent elements of S . In this paper, we investigate the condition under f≤g which for two elements f,g in TsE ( x ) . Also, the maximal and minimal elements are described.