Injective positively ordered monoids. II
type: Item
instance of: scholarly article
title: Injective positively ordered monoids. II
zbMATH Open document ID: 0790.06017
DOI: 10.1016/0022-4049(92)90105-O
author: Friedrich Wehrung
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: This is a continuation of work on positively ordered monoids (POM) began by the author in the paper reviewed above, where the author gave arithmetical characterizations of injective POM's. Here, criteria are found for embedding an arbitrary POM into an injective POM, in which case the POM is called regular. Unlike the case in Abelian groups or Boolean algebras, many POM's are not regular. The most fundamental regular POM is the extended real line \(P\), and injective POM's are exactly the retracts of powers of \(P\). A class of POM's larger than the injective POM's is refinement POM's. Statements proved by \textit{A. Tarski} [Cardinal algebras (1949; Zbl 0041.345)] for cardinal algebras are shown to be ``approximately true'' in all refinement POM's. A functor from the category of refinement POM's to itself allows characterization of the evaluation map of a refinement POM with applications to equidecomposability type POM's, a regularity criterion, and the existence of Banach limits.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 166153
zbMATH Keywords: positively ordered monoids
zbMATH Keywords: extended real line
zbMATH Keywords: cardinal algebras
zbMATH Keywords: category
zbMATH Keywords: equidecomposability type
zbMATH Keywords: regularity
zbMATH Keywords: Banach limits
Wikidata QID: Q114215374
reviewed by: Bernard Madison
MaRDI profile type: Publication
cites work: Q5331549
cites work: Q4169863
cites work: Cardinal addition and the axiom of choice
cites work: Model theory
cites work: Q5181909
cites work: Cardinal Algebras and Measures Invariant under Equivalence Relations
cites work: Cardinal algebras of functions and integration
cites work: On the Representation of Cardinal Algebras by Directed Sums
cites work: Q5583934
cites work: Group Representations and Cardinal Algebras
cites work: Q3040203
cites work: Refinement monoids, Vaught monoids, and Boolean algebras
cites work: Banach-Tarski Decompositions Using Sets with the Property of Baire
cites work: An Archimedean property of cardinal algebras
cites work: The Dimension Theory of Certain Cardinal Algebras
cites work: On groups with full Banach mean value
cites work: The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set
cites work: Q3435573
cites work: Q5562332
cites work: Q5726943
cites work: On some fundamental problems concerning isomorphism of Boolean algebras
cites work: The axiom of choice
cites work: Q3852172
cites work: A solution of a problem of R. Sikorski
cites work: Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem
cites work: The Banach-Tarski paradox for the hyperbolic plane
cites work: Folner's Conditions for Amenable Semi-Groups.
cites work: Q4404885
cites work: Definability of measures and ultrafilters
cites work: Q4027472
cites work: Weak Cardinal Algebras
cites work: Duality for Cardinal Algebras
cites work: Q5802110
cites work: Q3687182
cites work: Q5752784
cites work: Metric properties of positively ordered monoids
cites work: Injective positively ordered monoids. I
Recommended article: Injective positively ordered monoids. I
Recommended article: Injective hulls of partially ordered monoids
Recommended article: Classification of monoids by injectivities. II: CC-injectivity.
Recommended article: Restricted injectivity, transfer property and decompositions of separative positively ordered monoids.
Recommended article: Q4263818
Recommended article: Classification of monoids by injectivities. I: C-injectivity.
Recommended article: Injectivity of \(S\)-posets with respect to down closed regular monomorphisms.
Recommended article: Injective hulls of monars over inverse semigroups.
Recommended article: An approach to injective acts over monoids based on indecomposability
Recommended article: Q3739172
description: scientific article; zbMATH DE number 166153