On groups of hypersubstitutions
type: Item
instance of: scholarly article
title: On groups of hypersubstitutions
zbMATH Open document ID: 1216.08001
author: J. D. H. Smith
published in: Algebra Universalis
publication date: 2011-01-07T00:00:00Z
review text: Let \(\tau : \Omega \to \mathbb{N}\) be a finitary type, where \(\Omega\) is a set of symbols of algebraic operations and \(\tau (\omega )\) is the arity of the \(\omega\). Let \(P\) denote the ordered set of positive integers and \(P\Omega \) denote the set of \(\Omega\)-words over \(P\). The so-called derived type \(\tau': P\Omega \to \mathbb{N}\) was defined in [\textit{J. D. H. Smith} and \textit{A. B. Romanowska}, Post-modern algebra. New York, NY: Wiley (1999; Zbl 0946.00001), \S IV 1.3]. A hypersubstitution of type \(\tau \) is a morphism \(\sigma : \tau \to \tau '\) in the slice category \(\text{Set}/N\), that is, a function \(\sigma : \Omega \to P\Omega \) such that \(\tau ' \circ \sigma =\tau\). Each hypersubstitution \(\sigma : \Omega \to P\Omega \) extends to a corresponding function \(\hat{\sigma} : P\Omega \to P\Omega \). Let \(\mathcal V\) be a variety of algebras of type \(\tau\). A hypersubstitutions \(\sigma\) is called (\(\mathcal V\)-)proper if \(\hat{\sigma} (u)=\hat{\sigma} (v) \) for every identity \(u=v\) of \(\mathcal V\). In the paper under review, the proper hypersubstitutions of a variety \(\mathcal V\) of the kind \(\sigma : \Omega \to \Omega \) are considered. If hypersubstitution \(\sigma : \Omega \to \Omega \) is invertible and (\(\mathcal V \)-)proper, it is called a \(\mathcal V\)-hyperequivalence. In particular, it is proved that every monoid may be realized as a monoid of hypersubstitutions and every automorphism group of a monoid is isomorphic to the \(\mathcal V\)-hyperequivalence group of some variety \(\mathcal V\). Since groups of hyperequivalences can be regarded as symmetries of algebraic theories, the author takes an interest in the phenomenon of symmetry-breaking. He considers lattice theory and quasigroup theory from this point of view.
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 5833772
zbMATH Keywords: hypersubstitution
zbMATH Keywords: lattice
zbMATH Keywords: one-way function
zbMATH Keywords: quasigroup
zbMATH Keywords: hyperquasigroup
zbMATH Keywords: loop transversal
zbMATH Keywords: Catalan loop
reviewed by: Grigori I. Zhitomirski
MaRDI profile type: Publication
full work available at URL: s00012-010-0087-y
OpenAlex ID: W2079532654
cites work: Q5507586
cites work: Q3979090
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cites work: Catalan loops
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DOI: 10.1007/S00012-010-0087-Y
description: scientific article; zbMATH DE number 5833772