Powers of subsets in a finite semigroup
type: Item
instance of: scholarly article
title: Powers of subsets in a finite semigroup
zbMATH Open document ID: 0833.20074
DOI: 10.1007/BF02573616
published in: Semigroup Forum
publication date: 1996-03-12T00:00:00Z
full work available at URL: https://eudml.org/doc/135396
review text: 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.
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 784127
zbMATH Keywords: finite semigroups
zbMATH Keywords: power semigroups
zbMATH Keywords: idempotent elements
author: Štefan Schwarz
reviewed by: Takayuki Tamura
MaRDI profile type: Publication
cites work: Q3848243
cites work: Q4132170
cites work: Finite globally-idempotent semigroups
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cites work: On the sum of two sets in a group
cites work: Subgroups of the Power Semigroup of a Finite Semigroup
cites work: Isomorphism problem of power semigroups of completely 0-simple semigroups
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description: scientific article; zbMATH DE number 784127