Banaschewski's theorem for \(S\)-posets: regular injectivity and completeness.
type: Item
instance of: scholarly article
title: Banaschewski's theorem for \(S\)-posets: regular injectivity and completeness.
zbMATH Open document ID: 1198.20060
author: Mojgan Mahmoudi
author: Hamid Rasouli
author: M. Mehdi Ebrahimi
published in: Semigroup Forum
publication date: 2010-05-21T00:00:00Z
review text: From the authors' abstract with some additions: In this paper we study the notion of injectivity in the category \(\mathbf{Pos}\text{-}\mathbf S\) of \(S\)-posets for a pomonoid \(S\). First we see that, although there is no non-trivial injective \(S\)-poset with respect to monomorphisms, \(\mathbf{Pos}\text{-}\mathbf S\) has enough (regular) injectives with respect to regular monomorphisms (sub \(S\)-posets). Then, recalling Banaschewski's theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for \(S\)-posets. We see for example, that regular injective \(S\)-posets are exactly the retracts of cofree \(S\)-posets over complete posets. The authors get some homological classification of pomonoids and pogroups like the following. All complete \(S\)-posets, over a pomonoid \(S\) where the identity is maximal, are regular injective iff \(S\) is a pogroup. If the identity is a top, then this is the case iff \(S=1\). Recall that regular monomorphisms are equalizers which in the present case are exactly order-embeddings.
reviewed by: Ulrich Knauer
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 5710597
zbMATH Keywords: \(S\)-posets
zbMATH Keywords: regular injectivity
zbMATH Keywords: completeness
zbMATH Keywords: pomonoids
zbMATH Keywords: injective posets
zbMATH Keywords: monomorphisms
zbMATH Keywords: pogroups
zbMATH Keywords: retracts
MaRDI profile type: Publication
full work available at URL: s00233-010-9207-4
OpenAlex ID: W2054571357
Wikidata QID: Q122661056
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DOI: 10.1007/S00233-010-9207-4
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description: scientific article; zbMATH DE number 5710597