### Contextual partial commutations

*Christian Choffrut, Robert George Mercaş*

#### Abstract

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}^{*}.Full Text: PDF PostScript