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.
Highlights
Later in 1986, the natural partial order on a regular semigroup was further extended to any semigroup S by Mitsch [3] as follows: a ≤ b iff a = xb = by, (2)
Partial transformation semigroup is the collection of functions from a subset of X into X with composition which is denoted by P(X)
In [8], the authors proved that Fix(X, Y) is a regular subsemigroup of T(X)
Summary
In 1986, Kowol and Mitsch [4] characterized the natural partial order on T(X) in terms of images and kernels They proved that an element α ∈ T(X) is maximal with respect to the natural order if and only if α is surjective or injective; α is minimal if and only if α is a constant map. They described lower and upper bounds for two transformations and gave necessary and sufficient conditions for their existence. We find the number of upper covers of minimal elements and the number of lower covers of maximal elements
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
