Monoides et semi-anneaux complets. (Complete monoids and semirings)
type: Item
instance of: scholarly article
title: Monoides et semi-anneaux complets. (Complete monoids and semirings)
zbMATH Open document ID: 0636.16019
DOI: 10.1007/BF02575025
author: Daniel Krob
published in: Semigroup Forum
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134905
review text: Let \((M,+)\) be a commutative monoid and assume that \(\sum_ Ix_ i\in M\) is defined for each family \((x_ i|\) \(i\in I)\) of elements \(x_ i\in M\), where I is any set [any at most countably finite set]. Then \((M,+,\sum)\) is called a complete [c-complete] monoid iff \(\sum\) extends all finite sums already defined in \((M,+)\) and (P)\(\sum_ Ix_ i=\sum_ J(\sum_{I_ j}x_ i)\) holds for each partition \(I=\cup (I_ j|\) \(j\in J)\) of I. Correspondingly, let \((M,+,\cdot)\) be a semiring which need not have an identity, but an additive neutral 0 which is multiplicatively absorbing. Then \((M,+,\cdot,\sum)\) is called a complete [c-complete] semiring iff \((M,+,\sum)\) is a complete [c- complete] monoid and z(\(\sum_ ix_ i)=\sum_ i(zx_ i)\) as well as \((\sum_ Ix_ i)z=\sum_ I(x_ iz)\) are satisfied for all \(z\in M\) and all families \((x_ i|\) \(i\in I)\) under consideration. Moreover, a suitable topology \({\mathcal T}\) on \((M,+)\) can be used to define \(\sum_ Ix_ i\) such that \((M,+,\sum)\) is a complete [c-complete] monoid, then called t-complete [tc-complete], and likewise for semirings \((M,+,\cdot).\) Various statements concerning these eight concepts and corresponding (counter-)examples are given. For instance, complete [c-complete] does not imply t-complete [tc-complete], each complete monoid has an additively absorbing element, and each tc-complete monoid (semiring) has a natural partial order. Referee's remarks: 1) The paper refers to \textit{S. Eilenberg} [Automata, Languages and Machines (1974; Zbl 0317.94045)] and \textit{W. Kuich} [Lect. Notes Comput. Sci. 267, 212-225 (1987; Zbl 0625.16026)]. But there and elsewhere (cf. below) complete [c-complete] has another meaning, since instead of the axiom (P) above a stronger one, say (GP), is assumed where partitions (all \(I_ j\neq \emptyset)\) are replaced by generalized partitions \((I_ j=\emptyset\) may occur). In fact, t-complete [tc- complete] monoids and semirings satisfy (GP) as well. 2) There ar…
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 4036023
zbMATH Keywords: countable complete semirings
zbMATH Keywords: infinite sums defined by topologies
zbMATH Keywords: commutative monoid
zbMATH Keywords: complete monoid
zbMATH Keywords: tc-complete monoid
reviewed by: Hanns Joachim Weinert
MaRDI profile type: Publication
OpenAlex ID: W2086115527
Recommended article: Monoides et semi-anneaux continus. (Continuous monoids and semirings)
Recommended article: COMPLETIONS OF MONOIDS WITH APPLICATIONS TO THE CUNTZ SEMIGROUP
Recommended article: On semifir monoid rings
Recommended article: Q4284203
Recommended article: Q4006421
Recommended article: On seminormal monoid rings
Recommended article: Q3808145
Recommended article: Continuous monoids and semirings
Recommended article: Q4294065
Recommended article: On a class of residuated semilattice monoids
description: scientific article; zbMATH DE number 4036023