Near unanimity: An obstacle to general duality theory
type: Item
instance of: scholarly article
title: Near unanimity: An obstacle to general duality theory
zbMATH Open document ID: 0824.08007
DOI: 10.1007/BF01190710
author: Brian A. Davey
author: Ralph McKenzie
author: Lutz Heindorf
published in: Algebra Universalis
publication date: 1995-11-09T00:00:00Z
review text: Let \(\underline P= (P; F)\) be an algebra. Let \(\underset\widetilde{} P= (P; G, H, R, T)\) be a structure such that all \(n\)-ary (partial) operations from \(G\) \((H)\) are homomorphisms of the \(n\)th power of \(\underline P\) (of domain) into \(\underline P\), all \(n\)-ary relations from \(R\) are subuniverses of \(\underline P\), and \(T\) is the discrete topology on \(P\). Let \({\mathcal A}= \text{ISP}(\underline P)\) and \({\mathcal X}= \text{IS}_{\text{c}} \text{P}(P)\) (\(\text{S}_{\text{c}}\) means closed substructures). There are well-defined hom-functors \(D: {\mathcal A}\to {\mathcal X}\) and \(E: {\mathcal X}\to {\mathcal A}\). If \(ED(\underline A)\) is an isomorphism for all \(\underline A\in {\mathcal A}\), we say that \(\underline P\) admits natural duality (\(\underset\widetilde{} P\) yields a natural duality) in the sense of B. A. Davey. It was proved by B. A. Davey and H. Werner that if \(\underline P\) has a term (near- unanimity term) of arity \(\geq 3\) such that \(T(x,\dots, x, y)= T(x,\dots, y, x)= T(x, y, \dots, x)= T(y, \dots, x)\) are identities in \(\underline P\), then \(\underline P\) admits a natural duality. The authors prove that in the finite case \(\underline P\) admits a natural duality and has a distributive lattice of congruences iff \(\underline P\) has a near-unanimity term iff \(\underline P\) has a distributive lattice of congruences and for every compact totally disconnected topological space \(X\) each subalgebra of the algebra \(C(X, \underline P)\) of all continuous functions is closed in the pointwise topology. Some other connections between these notions are given.
Mathematics Subject Classification ID: 08A99
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 759392
zbMATH Keywords: finite algebra
zbMATH Keywords: near-unanimity term
zbMATH Keywords: algebra of continuous functions
zbMATH Keywords: closed substructures
zbMATH Keywords: hom-functors
zbMATH Keywords: natural duality
zbMATH Keywords: congruences
zbMATH Keywords: topology
reviewed by: Ernest Płonka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 759392