Representations of the free profinite object over DA

In this paper, we extend to DA some techniques developed by Almeida and Weil, and Almeida and Zeitoun for the pseudovariety R to obtain representations of the implicit operations on DA: by labeled trees of finite height, by quasi-ternary labeled trees, and by labeled linear orderings. We prove that two implicit operations are equal over DA if and only if they have the same representation, for any of the three representations. We end the paper by relating these representations.info:eu-repo/semantics/publishedVersio

Reducibility of pointlike problems

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all fi nite semigroups in which the order of every subgroup is a product of elements of a fi xed set of primes; the pseudovariety of all fi nite semigroups in which every regular J-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain omega-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (A) and of all finite semigroups in which all regular elements are idempotents (DA).ANR 2010 BLAN 0202 01 FRE

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.The work of the first, third, and fourth authors was partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first three authors was also partially supported respectively by CMUP (UID/MAT/ 00144/2019), CMUC (UID/MAT/00324/2019), and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The work of the fourth author was partly supported by ANR 2010 BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-000