Functions for the General Solution of Parametric Word Equations

: In this article we introduce the functions Fi (x 1 , x 2 ) l1,..., ls and Th (x 1 , x 2 , x 3 ) i l1,..., l2s (i = 1, 2, 3), of the word variables x i and of the natural number variables li, where s ³ 0. By means of these functions, we give exactly the general solution (i.e. the set of all the solutions) of the first basic parametric equation: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , in a free monoid. 1. Introduction The following four parametric equations: x 1 x 2 x 3 x 4 = x 3 x 1 l x 2 x 5 , x 1 x 2 x 3 x 4 = x 2 x 3 l x 1 x 5 , x 1 x 2 2 x 3 x 4 = x 3 x 1 2 x 2 x 5 , x 1 x 2 l+1 x 3 x 4 = x 3 x 2 µ+1 x 1 x 5 , in a free monoid, are called basic equations. They arise in the graph of the prefixeequations in free monoid (cf. [2], [3]) and play an important role in the hierarchy of the parametric equations, in reason of the structures of their solutions. In particular, the general solution of any equation in a free monoid of the form F(x 1 , x 2 , x 3 ) x 4 = Y(x 1 , x 2 , ..