Infinite-vertex free profinite semigroupoids and symbolic dynamics. (Q1004467)

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 \(A\) is finite. A symbolic dynamical system of \(A^\mathbb{Z}\) is a nonempty closed subset \(X\) of \(A^\mathbb{Z}\) that contains the orbits of its elements. Symbolic dynamical systems are also called shift spaces or subshifts. The authors bring together symbolic dynamics and relatively free profinite semigroupoids. Some fundamental questions about infinite-vertex (free) profinite semigroupoids are clarified, putting in evidence differences with the finite-vertex case. This is done with examples of free profinite semigroupoids generated by the graph of a subshift. It is also proved that for minimal subshifts, the infinite edges of such free profinite semigroupoids form a connected compact groupoid.