Powers of subsets in a finite semigroup
Let \(S\) be a finite semigroup, \(|S|= n\). By the power semigroup \({\mathcal P} (S)\) of a semigroup \(S\) we mean \(\{A \mid A\subset S,\) \(A \neq \emptyset\}\) with the multiplication \(A \cdot B = \{ab \mid a \in A,\;b \in B\}\). Let \(P\) be a non-empty subset of \(S\) i.e. \(P \in {\mathcal P} (S)\). It is obvious that \(\{P, P^2, \dots\}\) contains only a finite number of elements (i.e. subsets of \(S\)). Let \(k = k(P)\) denote the least integer for which \(P^k = P^t\) for some \(t > k\), and let \(d = d(P) \geq 1\) be the least integer for which \(P^k = P^{k + d}\). The sequence is \(P, P^2, \dots, P^{k - 1} |P^k, \dots, P^{k + d - 1} |P^k, \dots\) where \(\{P^k, \dots, P^{k + d - 1}\}\) is the periodic part and a cyclic group of order \(d\). The purpose of this paper is to find estimates i.e. upper bounds for \(k(P)\) in terms of \(n = |S|\). Two main theorems are presented. One of these treats the case where all idempotent elements of \(S\) commute with all elements of \(S\). In this case, for any \(P \in {\mathcal P} (S)\), \(k(P) \leq n\) where \(n = |S|\). This is the best estimate. Another result discusses the general case. If \(|S|= n\), then \(k(P) \leq 2n - 2\). With respect to \(d(P)\), it is proved that a polynomial estimate is impossible. As the author indicates, the study of \(k(P)\) and \(d(P)\) has not been done by other semigroupists. This paper is highly evaluated. Also the paper is kindly written such that readers can easily understand, in particular many examples are given.
scientific article; zbMATH DE number 784127
scientific article