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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 \(|b\varepsilon^{-1}|\geq 3\).

20M20 Semigroups of transformations, relations, partitions, etc.