Infinite-vertex free profinite semigroupoids and symbolic dynamics

Abstract

Communicated by M. Sapir MSC:Primary: 20M05 20M07 secondary: 37B10 20M50 a b s t r a c t 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.

Iterated periodicity over finite aperiodic semigroups

“…Since the set of factors of an element of a compact semigroup is always closed, the equivalence of conditions (a), (b), and (c) is the statement of [4, Theorem 2.11] for V = S and one can check that the results of [4, Section 2], with the same proofs, apply to any pseudovariety V containing LSl (see also [6,Section 6] for an alternative approach). We proceed to establish the equivalence with the remaining conditions.…”

“…See [31] for the definition of pseudovarieties of categories and their relevance for the theory of pseudovarieties of finite semigroups. See also [6] for caveats and pitfalls that one should be aware of when handling pseudovarieties of categories, particularly when profinite techniques are involved.…”

A combinatorial property of ideals in free profinite monoids

“…Alfredo Costa has pointed out to me that the theorem can also be deduced via an inductive argument using [2,Lemma 4.8]; however, that lemma is implicit in [6] and can easily be proved by the technique employed in [6] and here. In fact, our main theorem holds for any A-generated profinite monoid whose finite images are closed under the Henckell-Schützenberger expansion.…”

“…The author would like to thank Alfredo Costa for pointing out that the ideals in the main theorem need not be assumed closed, as well as for reference [2]. The author was supported in part by NSERC.…”

Semidirect product with an order-computable pseudovariety and tameness

“…The profinite approach was added in [19,13], with relatively free profinite semigroupoids being generated respectively by finite and profinite graphs. Later, it was shown that the case of graphs with infinite vertex sets is rather delicate [8], the treatment in [13] containing some errors. As we require very little on this topic, we will not go any further into it, referring the reader to the above references for details.…”