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... ALFREDO COSTA 2. The free profinite semigroupoid generated by ∂XP For general background on pseudovarieties of semigroups and of semigroupoids, and on their relatively free profinite structures, see =-=[2, 1, 8]-=-. We also adopt the notation of [4], which we recall here for the reader’s benefit. Let A be a finite alphabet and let V be a pseudovariety of semigroups. The free pro-V semigroup generated by A is de...

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... ALFREDO COSTA 2. The free profinite semigroupoid generated by ∂XP For general background on pseudovarieties of semigroups and of semigroupoids, and on their relatively free profinite structures, see =-=[2, 1, 8]-=-. We also adopt the notation of [4], which we recall here for the reader’s benefit. Let A be a finite alphabet and let V be a pseudovariety of semigroups. The free pro-V semigroup generated by A is de...

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...finite-vertex case, the topological closure of the subsemigroupoid generated by the graph may not be a subsemigroupoid of the corresponding free profinite semigroupoid. This problem was overlooked in =-=[5]-=- and [4], where infinite-vertex free profinite semigroupoids are seriously considered for the first time as a tool to be used in the study of relatively free profinite semigroups. While their role in ...

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...groups, pseudovarieties, free profinite semigroups, free profinite semigroupoids, profinite graphs. AMS Subject Classification (2010): primary 20M07; secondary 20M17, 20M05, 22A15. 1. Introduction In =-=[6]-=- one finds a systematic study of relatively free profinite categories and relatively free profinite semigroupoids generated by profinite graphs with a finite number of vertices. The case of profinite ...

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... For the case where W is the pseudovariety Sd of all finite semigroupoids, one uses “profinite” as a synonym of “pro-W”. We remark that there is an unpublished example due to G. Bergman (mentioned in =-=[7]-=-) of an infinite-vertex semigroupoid which is profinite according to this definition, but which is not an inverse limit of finite semigroupoids. Let Γ be a profinite graph and let W be a pseudovariety...

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... ALFREDO COSTA 2. The free profinite semigroupoid generated by ∂XP For general background on pseudovarieties of semigroups and of semigroupoids, and on their relatively free profinite structures, see =-=[2, 1, 8]-=-. We also adopt the notation of [4], which we recall here for the reader’s benefit. Let A be a finite alphabet and let V be a pseudovariety of semigroups. The free pro-V semigroup generated by A is de...

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...ively free profinite semigroupoids generated by profinite graphs with a finite number of vertices. The case of profinite graphs with an infinite number of vertices is more delicate, as highlighted in =-=[3]-=-. The main problem is that in the infinite-vertex case, the topological closure of the subsemigroupoid generated by the graph may not be a subsemigroupoid of the corresponding free profinite semigroup...

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...y subset X of A with at least two elements. Proof : Since A is finite, ΩAV is metrizable, and therefore so is ∂XΩAV. The relatively free profinite semigroupoids which intervene in the main results in =-=[4]-=- are of the form Ω∂XΩAVW. We remark that in (the applications of) Theorem 2.5 of [4], the graph ∂XΩAV is identified with the graph ∂X(A +) and the subsemigroupoid 〈∂XΩAV〉 of Ω∂XΩAVW generated by ∂XΩAV...

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...gical graphs, with finite graphs having the discrete topology. It is folklore that the profinite graphs are precisely the topological graphs whose topology is a Boolean space (a proof can be found in =-=[9]-=-.) It follows immediately from this characterization that the graph ∂XP is profinite if P is a closed subset of ΩAV, which happens in particular when P = ΩXV, the case considered in the main results o...