On classification of monoids by properties of cofree acts
type: Item
instance of: scholarly article
title: On classification of monoids by properties of cofree acts
zbMATH Open document ID: 0944.20050
author: Valdis Laan
published in: Semigroup Forum
publication date: 2000-06-21T00:00:00Z
review text: In the category \(\mathbf{Act}\)-\(S\) of right \(S\)-acts \(|I|\)-cofree acts are of the form \(I^S\) with the usual action (from the right), \(I\) a set. The author proves, for example, the equivalence of the following conditions: (1) All principally weakly injective \(S\)-acts satisfy condition \((E)\), (2) All weakly injective \(S\)-acts satisfy condition \((E)\), (3) All injective \(S\)-acts satisfy condition \((E)\), (4) All cofree \(S\)-acts satisfy condition \((E)\), (5) Every right congruence \(\varrho(s,t)\) is the kernel of some left translation, and similar results where condition \((E)\) is replaced by condition \((P)\) or by ``strongly flat'', where for commutative monoids the assertion corresponding to (4) is equivalent to \(S\) being a group. If condition \((E)\) is replaced by ``projective'', then the assertion corresponding to (4) is equivalent to \(S=\{1\}\). In the last section implications of the form ``divisible'' \(\Longrightarrow (E)/(P)/\)``strongly flat'' are used to characterize \(S=\{1\}\) or \(S=\{1,0\}\)/\(S\) a group/\(S=\{1\}\) in this order.
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 1330653
zbMATH Keywords: monoids
zbMATH Keywords: categories of right acts
zbMATH Keywords: cofree acts
zbMATH Keywords: injective acts
zbMATH Keywords: flat acts
zbMATH Keywords: projective acts
reviewed by: Ulrich Knauer
MaRDI profile type: Publication
full work available at URL: s002339900034
OpenAlex ID: W1974973505
DOI: 10.1007/S002339900034
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description: scientific article; zbMATH DE number 1330653