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Semigroups of transformations commuting with idempotents. (English) Zbl 1005.20046
Let $$X^0_n=\{0,1,2,\dots,n\}$$ and let $$T^0_n$$ be the semigroup of all full transformations on $$X^0_n$$ which fix $$0$$. It is well known that $$T^0_n$$ is isomorphic to the semigroup of all partial transformations on a set of $$n$$ elements and this is the reason for the author’s interest in $$T^0_n$$. Let $$\varepsilon$$ be an idempotent in $$T^0_n$$ and let $$C(\varepsilon)$$ be the subsemigroup of $$T^0_n$$ which consists of all those elements which commute with $$\varepsilon$$. He determines Green’s relations for $$C(\varepsilon)$$ and he characterizes the regular elements of $$C(\varepsilon)$$. In particular, he gives necessary and sufficient conditions for $$C(\varepsilon)$$ to be a regular semigroup. Specifically, he shows that $$C(\varepsilon)$$ is a regular semigroup if and only if either (1) $$x\varepsilon=y\varepsilon\neq 0$$ implies $$x=y$$ or (2) there do not exist two distinct fixed points $$a$$ and $$b$$ of $$\varepsilon$$ such that $$|a\varepsilon^{-1}|\geq 2$$ and $$|b\varepsilon^{-1}|\geq 3$$.

##### MSC:
 20M20 Semigroups of transformations, relations, partitions, etc.