On clones of operations over finite Boolean algebras
type: Item
instance of: scholarly article
title: On clones of operations over finite Boolean algebras
zbMATH Open document ID: 1160.08002
author: Dragan Mašulović
published in: Algebra Universalis
publication date: 2009-02-17T00:00:00Z
review text: Let \(\mathcal{O}_{A}=\bigcup_{n\geq 1}A^{A^{n}}\) denote the set of all finitary operations on \(A\). A set \(\mathcal{C}\subseteq \mathcal{O}_{A}\) of finitary operations is a clone of operations on \(A\) if it contains all the projection maps \(\pi_{i}^{n}:A^{n}\to A,\langle x_{1},\dots,x_{n}\rangle\mapsto x_{i}\), and is closed with respect to superposition of operations in the following sense: for all \(f,g_{1},\dots,g_{n}\in \mathcal{C}\) such that \(\text{ar}(f)=n\) and \(\text{ar}(g_{1})=\dots=\text{ar}(g_{n})\) we have \(f(g_{1},\dots,g_{n})\in \mathcal{C}\). For every set \(F\subseteq \mathcal{O}_{A}\) there is a least clone containing \(F\). It is referred to as the clone generated by \(F\) and denoted by Clo(\(F\)). For a finite Boolean algebra \(\mathbf{B}\) let \(\mathcal{O}_{\mathbf{B}}=\bigcup_{n\geq 1}\text{hom}(\mathbf{B}^{n},\mathbf{B})\) denote the set of all finitary operations over \(\mathbf{B}\). The set \(\mathcal{O}_{\mathbf{B}}\) is a clone of operations in a usual sense. A clone \(\mathcal{C}\subseteq \mathcal{O}_{\mathbf{B}}\) will be referred to as a clone of operations over \(\mathbf{B}\). Let \(\mathcal{L}_{\mathbf{B}}\) denote the lattice of clones of operations over \(\mathbf{B}\). A dual operation over a finite set \(X\) is any mapping \(f:X\to m\cdot X\), where \(m\cdot X=\{1,\dots,m\}\times X\) is the \(m\)th copower of \(X\). We say that \(m\) is the \textit{arity} of \(f\) and write \(\text{ar}(f)=m\). Every dual operation \(f:X\to m\cdot X\) gives rise to two maps, val\(f:X\to X\), the value of \(f\), and lbl\(f:X\to \{1,\dots,m\}\), the labelling of \(f\), defined by \(f(x)=\langle j,y\rangle\) if and only if val\(f(x)=y\) and lbl\(f(x)=j\). Let \(\mathcal{\overline{O}}_{X}=\bigcup_{n\geq 1}(n\cdot X)^{X}\) denote the set of all finitary dual operations over \(X\), and \(\mathcal{\overline{J}}_{X}\) the set of all injection maps \(\iota_{i}^{n}:X\to n\cdot X, x\mapsto \langle i,x\rangle, n\in \mathbf{N}, 1\leq i\leq n\). A set…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 5508759
zbMATH Keywords: clones
zbMATH Keywords: Boolean algebras
zbMATH Keywords: dual operations
zbMATH Keywords: automorphisms
zbMATH Keywords: congruences
MaRDI profile type: Publication
full work available at URL: s00012-008-2118-5
OpenAlex ID: W2145558985
DOI: 10.1007/S00012-008-2118-5
description: scientific article; zbMATH DE number 5508759