Locally closed semirings
type: Item
instance of: scholarly article
title: Locally closed semirings
zbMATH Open document ID: 1015.16052
published in: Monatshefte für Mathematik
publication date: 2002-12-01T00:00:00Z
review text: A \(^*\)-semiring is an additively commutative semiring \((S,+,\cdot)\) with absorbing zero and identity 1 equipped with a star operation \(^*\colon S\to S\). If \((x+y)^*=(x^*y)^*x^*\) and \((xy)^*=1+x(yx)^*y\) for all \(x,y\in S\) then \((S,+,\cdot)\) is called a Conway semiring. An iteration semiring is a Conway semiring satisfying all so-called group-equations. Now, let \((S,+,\cdot)\) be a locally closed semiring, i.e., for every \(a\in S\) there is some integer \(k\geq 0\) such that \(1+a+\cdots+a^k=1+a+\cdots+a^{k+1}\). Then \((S,+,\cdot)\) becomes a star semiring by the star operation \(a^*=1+a+\cdots+a^k\). It is proved that every locally closed and commutative semiring is an iteration semiring satisfying \(1^*=(1^*)^*\).
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 68Q70
zbMATH DE Number: 1837281
zbMATH Keywords: commutative semirings
zbMATH Keywords: Conway semirings
zbMATH Keywords: iteration semirings
zbMATH Keywords: locally closed semirings
author: Zoltán Ésik
author: W. Kuich
reviewed by: Udo Hebisch
MaRDI profile type: Publication
full work available at URL: s00605-001-0481-9
OpenAlex ID: W2061736301
DOI: 10.1007/S00605-001-0481-9
description: scientific article; zbMATH DE number 1837281