Natural Partial Orders on Transformation Semigroups with Fixed Sets

Abstract

LetXbe a nonempty set. For a fixed subsetYofX, letFixX,Ybe the set of all self-maps onXwhich fix all elements inY. ThenFixX,Yis a regular monoid under the composition of maps. In this paper, we characterize the natural partial order onFix(X,Y)and this result extends the result due to Kowol and Mitsch. Further, we find elements which are compatible and describe minimal and maximal elements.

On certain Semigroup of Order-decreasing Full Contraction Mappings of a Finite Chain

“…This relation is known to be the natural partial order on a semigroup S. If S is regular then α ≤ β if and only if α = ǫβ = βη for some ǫ, η ∈ E(S), and if S is a semilattice of idempotents then ǫ ≤ η if and only if ǫ = ǫη = ηǫ for some ǫ, η ∈ E(S). Partial order relations on various semigroups of the partial transformations have been investigated by many authors, see for example [6,18,21,20]. It is worth noting that the semigroup OCT n is not regular (see [32]).…”

On Connectedness and Completeness of Cayley Digraphs of Transformation Semigroups With Fixed Sets

“…Later in 2016, Chaiya, Honyam and Sanwong [2] presented the characterization of the natural partial order on Fix(X, Y ).…”