Abstract
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We consider the monoid T with the presentation < a, b; aab = aba > which is ''close'' to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid \a, b\* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in \a, b\*.
Highlights
Trace monoids are obtained from free monoids by allowing certain pairs of generators to commute, which is the reason why they are known as free partially commutative monoids
We investigate a natural extension by imposing on these partial commutations to be controlled by the context, e.g., we may specify that the letters a and b commute when preceded by the letter c but not by the letter d and call them contextual trace monoids, abbreviated as c-trace monoids
Apart from the trace monoids themselves, the plactic monoids originate from the rules of the jeu de taquin on a set of finite elements – the generators – and their relators consist of pairs of words of length 3, see Chapter 5 of [9]
Summary
Trace monoids are obtained from free monoids by allowing certain pairs of generators to commute, which is the reason why they are known as free partially commutative monoids. Apart from the trace monoids themselves, the plactic monoids originate from the rules of the jeu de taquin on a set of finite elements – the generators – and their relators consist of pairs of words of length 3, see Chapter 5 of [9]. We state a factorization result where Łukasiewicz words are involved, yielding a linear algorithm deciding the equivalence of two words; differently said, the word problem is linear for congruences generated by the relation aab = aba. This result is instrumental for solving equations and allows us to characterize the solutions of the elementary equations such as the commutation and conjugacy equations. As a last example of difference between ordinary and c-traces, we show that the product of two recognizable subsets of contextual traces need not be recognizable
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