Rigidity of unary algebras and its application to the \({\mathcal {HS} = \mathcal {SH}}\) problem
type: Item
instance of: scholarly article
title: Rigidity of unary algebras and its application to the \({\mathcal {HS} = \mathcal {SH}}\) problem
zbMATH Open document ID: 1221.08004
author: Tomasz Brengos
published in: Algebra Universalis
publication date: 2011-05-11T00:00:00Z
review text: \textit{H.~P. Gumm} and \textit{T. Schröder} [Algebra Univers. 53, No.~2--3, 229--252 (2005; Zbl 1086.08002)] showed that if a functor \(T:\) {Set} \(\to\) {Set} preserves preimages, then \({\mathcal{H}}{\mathcal{S}}(K)={\mathcal{S}}{\mathcal{H}}(K)\) is valid for any class \(K\) of \(T\)-algebras. The present author applies a construction of rigid unary algebras to prove that \({\mathcal{H}}{\mathcal{S}}\neq{\mathcal{S}}{\mathcal{H}}\) for a class of {Set}-endofunctors not preserving non-empty preimages. For related results cf. also papers by \textit{T. Brengos} and \textit{V. Trnková} [Algebra Univers. 63, No.~2--3, 283--301 (2010; Zbl 1220.03011)] and \textit{V. Trnková} [Commentat. Math. Univ. Carol. 10, 323--352 (1969; Zbl 0183.30401); ibid. 12, 143--174 (1971; Zbl 0232.18004)].
Mathematics Subject Classification ID: 08A60
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 08A70
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 18B05
zbMATH DE Number: 5886751
zbMATH Keywords: coalgebra
zbMATH Keywords: preimage preservation
zbMATH Keywords: functor
zbMATH Keywords: coalgebraic logic
reviewed by: Danica Jakubiková-Studenovská
MaRDI profile type: Publication
full work available at URL: s00012-011-0118-3
OpenAlex ID: W2023386710
cites work: The \({\mathcal {HS}} = {\mathcal {SH}}\) problem for coalgebras
cites work: Types and coalgebraic structure
cites work: Algebra and Coalgebra in Computer Science
cites work: Q3208855
cites work: Q5574731
cites work: Q5639842
cites work: Q2752015
DOI: 10.1007/S00012-011-0118-3
description: scientific article; zbMATH DE number 5886751