Injective positively ordered monoids. I
type: Item
instance of: scholarly article
title: Injective positively ordered monoids. I
zbMATH Open document ID: 0790.06016
DOI: 10.1016/0022-4049(92)90104-N
author: Friedrich Wehrung
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: A positively ordered monoid is a structure \((A,+,0,\leq)\), where \((A,+,0)\) is a commutative monoid and \(\leq\) is a partial preordering of \(A\) in which every element is positive and the ordering is compatible with \(+\). This work defines various kinds of complete positively ordered monoids which are contexts for studying measurable function spaces, equidecomposability types of spaces, partially ordered Abelian groups, and cardinal algebras. The strongest notion of complete, termed injective, is arithmetically determined to be complete with the added requirement of divisibility by all nonzero natural numbers. The motivation for many of the results lies in the work of \textit{A. Tarski} [Fundam. Math. 31, 47-66 (1938; Zbl 0019.05403); Cardinal algebras (1949; Zbl 0041.345)] on the existence of a homomorphism from a commutative monoid to the extended positive real line and on cardinal algebras. The first result, the injectivity of the extended positive real line, is framed in the category of positively ordered monoids, and injective positively ordered monoids are posed as an algebraic enrichment of cardinal algebras.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 166152
zbMATH Keywords: positively ordered monoid
zbMATH Keywords: measurable function spaces
zbMATH Keywords: equidecomposability types
zbMATH Keywords: partially ordered Abelian groups
zbMATH Keywords: cardinal algebras
zbMATH Keywords: extended positive real line
zbMATH Keywords: category
Wikidata QID: Q114215377
reviewed by: Bernard Madison
MaRDI profile type: Publication
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full work available at URL: 0022-4049(92)90104-n
OpenAlex ID: W4205373660
description: scientific article; zbMATH DE number 166152