Locally closed semirings and iteration semirings.
type: Item
instance of: scholarly article
title: Locally closed semirings and iteration semirings.
zbMATH Open document ID: 1072.16041
author: Xianzhong Zhao
published in: Monatshefte für Mathematik
publication date: 2005-04-29T00:00:00Z
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 semiring is a Conway semiring. Moreover, if \((S,+,\cdot)\) is locally closed then, for any nonnegative integer \(n\), the matrix semiring \(S^{n\times n}\) is also locally closed. This implies that every locally closed semiring is an iteration semiring.
reviewed by: Udo Hebisch
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 68Q70
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 2162477
zbMATH Keywords: locally closed semirings
zbMATH Keywords: iteration semirings
zbMATH Keywords: Conway semirings
zbMATH Keywords: matrix semirings
zbMATH Keywords: polynomial semirings
MaRDI profile type: Publication
full work available at URL: s00605-004-0265-0
OpenAlex ID: W2127942730
DOI: 10.1007/S00605-004-0265-0
Recommended article: Locally closed semirings
Recommended article: On iteration semiring-semimodule pairs.
Recommended article: Iteration Semirings
Recommended article: ON SEMIREAL CLOSED RINGS
Recommended article: Q3682637
Recommended article: On semilocal rings
Recommended article: Q5718439
Recommended article: Q3009686
Recommended article: Q5489842
Recommended article: Q3096235
description: scientific article; zbMATH DE number 2162477