Abstract
Mitsch defined the natural partial order $$\le $$ on a semigroup S as follows: $$a\le b$$ if and only if $$a=xb=by, a=xa$$ for some $$x,y\in S^1$$ . Let $${{{\mathcal {T}}}}_X$$ be the full transformation semigroup on a finite set $$X=\{1,2,\ldots ,n\}$$ . Let $$\rho $$ be an equivalence relation on X and $$\preceq $$ be a total order on the partition set $$X/\rho $$ of X induced by $$\rho $$ . Denote by $${\overline{x}}$$ the $$\rho $$ -class containing $$x\in X$$ . In this paper, we endow the partition order-decreasing transformation subsemigroup of $${{{\mathcal {T}}}}_X$$ defined by $$\begin{aligned} T(\rho ,\preceq )=\{f\in {{{\mathcal {T}}}}_X: \forall \,\,x\in X, \overline{f(x)}\preceq {\overline{x}}\} \end{aligned}$$ with the natural partial order and give a characterization for this order. Then we determine the compatibility of their elements and find all the minimal and maximal elements.
Published Version
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