An algebraic closure for barycentric algebras and convex sets
type: Item
instance of: scholarly article
title: An algebraic closure for barycentric algebras and convex sets
zbMATH Open document ID: 1261.08002
author: Gábor Czédli
author: Anna Romanowska
published in: Algebra Universalis
publication date: 2012-11-27T00:00:00Z
review text: Let \(T\) be a subring of the real numbers containing the ring of integers. The open unit interval of \(T\) is then \(I^o(T):=\{r\in T:0<r<1\}\). For each \(p\in I^o(T)\), the authors define a so-called barycentric operation \(\underline{p}:T^2\rightarrow T\) by \(\underline{p}(x,y):=(1-p)x+py\). The members of the variety \(V(T)\) generated by the algebra \((T,\{\underline{p}:p\in I^o(T)\})\) are called (generalized) barycentric algebras. They are basic examples of idempotent and entropic algebras, called modes; see [\textit{A. B. Romanowska} and \textit{J. D. H. Smith}, Modes. Singapore: World Scientific (2002; Zbl 1012.08001)]. The main aim of this paper is to describe some algebraic closure of cancellative barycentric algebras. The latter generalize convex subsets of real affine space \(\mathbb{R}^n\). It is shown that a convex subset of \(\mathbb{R}^n\) is topologically closed iff it is closed in the authors' sense. They consider the following classes: \(Q(T)\), the quasivariety generated by \((T,\{\underline{p}:p\in I^o(T)\})\), \(C(T)\), the quasivariety of cancellative barycentric algebras, and \(H(T)\), some special quasivariety of barycentric algebras. Algebraic closures are constructed here for members of \(H(T)\). Obviously, \(Q(T)\subseteq C(T)\subseteq V(T)\). The main theorem shows that \(Q(T)\subseteq H(T)\subseteq C(T)\) and that each member of \(H(T)\) has a unique closure in \(H(T)\). It also shows when \(H(T)=C(T)\).
reviewed by: Anna Zamojska-Dzienio
Mathematics Subject Classification ID: 08A99
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 6110483
zbMATH Keywords: mode
zbMATH Keywords: barycentric algebra
zbMATH Keywords: medial groupoid
zbMATH Keywords: entropic groupoid
zbMATH Keywords: entropic algebra
zbMATH Keywords: convex set
zbMATH Keywords: algebraic closure
zbMATH Keywords: closed algebra
MaRDI profile type: Publication
full work available at URL: s00012-012-0195-y
OpenAlex ID: W2047484791
cites work: Q3934450
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DOI: 10.1007/S00012-012-0195-Y
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description: scientific article; zbMATH DE number 6110483