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