Infinite-vertex free profinite semigroupoids and symbolic dynamics.
type: Item
instance of: scholarly article
title: Infinite-vertex free profinite semigroupoids and symbolic dynamics.
zbMATH Open document ID: 1179.20050
author: Jorge Almeida
author: Alfredo Costa
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-10T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/10043
review text: We shall recall some definitions. By a graph the authors mean a directed multigraph, that is a disjoint union \(G=V_G\cup E_G\) of a set \(V_G\) of vertices with a nonempty set \(E_G\) of edges together with two incidence maps \(\alpha,\omega\) from \(E_G\) to \(V_G\). Two edges \(e\) and \(f\) are said to be consecutive if \(\omega(e)=\alpha(f)\). Let \(S\) be a graph. By \(D_S\) is denoted the set of pairs of consecutive edges of \(S\). They say that \(S\) is a semigroupoid if the set of edges of \(S\) is endowed with a partial binary operation ``\(\cdot\)'', such that: 1) given edges \(s\) and \(t\) of \(S\), the product \(s\cdot t\) is an edge which is defined if and only if \((s,t)\in D_S\); 2) if \((s,t)\in D_S\) then \(\alpha(s\cdot t)=\alpha(s)\) and \(\omega(s\cdot t)=\omega(t)\); 3) if \((s,t)\in D_S\) and \((t,r)\in D_S\) then \((s\cdot t)\cdot r=s\cdot(t\cdot r)\). Let \(\Gamma\) be a graph. The graph \(\Gamma^+\) is the graph whose vertices are those of \(\Gamma\) and whose edges from a vertex \(x\) to a vertex \(y\) are the paths of \(\Gamma\) from \(x\) to \(y\). Note that \(\Gamma\) is a subgraph of \(\Gamma^+\). Under the operation of concatenation of paths, \(\Gamma^+\) is the free semigroupoid generated by \(\Gamma\). Moreover, for a profinite graph \(\Gamma\) and for a pseudovariety \(\mathbf V\) of semigroupoids, the free pro-\(\mathbf V\)-semigroupoid generated by \(\Gamma\) is defined. Let \(A\) be the finite alphabet and let \(A^\mathbb{Z}\) be the set of sequences of letters of \(A\) indexed by \(\mathbb{Z}\). The shift in \(A^\mathbb{Z}\) is the bijective map \(\sigma\) from \(A^\mathbb{Z}\) to \(A^\mathbb{Z}\) defined by \(\sigma((x_i)_{i\in\mathbb{Z}})=(x_{i+1})_{i\in\mathbb{Z}}\). The orbit of \(x\in A^\mathbb{Z}\) is the set \(\mathcal O(x)=\{\sigma^k(x)\mid k\in\mathbb{Z}\}\). The set \(A^\mathbb{Z}\) is endowed with the product topology with respect to the discrete topology of \(A\). Note that \(A^\mathbb{Z}\) is compact, since \(A\) …
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 5527485
zbMATH Keywords: pseudovarieties of semigroupoids
zbMATH Keywords: relatively free profinite semigroupoids
zbMATH Keywords: symbolic dynamical systems
zbMATH Keywords: subshifts
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
OpenAlex ID: W2047703421
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DOI: 10.1016/J.JPAA.2008.08.009
description: scientific article; zbMATH DE number 5527485