Abstract
We characterize the solutions of all the equations in two unknowns in the free partially commutative monoids, and we show that the connected solutions are cyclic. Next, we define the transposition relation ( m and n are transposed iff m = xy and n = yx) and the conjugacy relation ( m and n are conjugate iff mλ = λn). We show that the conjugacy is the transitive closure of the transposition and that the set of conjugacy factors is a recognizable subset of the free partially commutative monoïd.
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