On systems of word equations with simple loop sets

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then u 1 and u 2 commute. Suppose x 0 u k+1 1 x 0 u k+1 1 v k+1 1 u 1 u 2 |v k+1 1 | < |x 0 | + |u k+2 1 |. This implies, together with (11), that |v k 1| < |x 0 u 1 |. Let d = v −k 1 x 0 u 1 . Note that d is a prefix of v 1 . If y 1 ≠ ǫ then v 1 and y 1 are not marked. Therefore y 1 = ǫ and d is comparable with v 2 . If |d| ≥ |v 2 | then v 1 and v 2 commute. d x 0 u 1 v k 1 v 1 v k 1 v 2 Suppose the contrary, which implies that d is a prefix of v 2 (as well as of v 1 ). Then both x 0 u k+2 1 and x 0 u k+1 1 u 2 are comparable with v1 k+1 d. Since |v k+1 1 d| = |x 0 u 1 | + |v 1 | > |x 0 u k+2 1 |, the words u 1 and u 2 are comparable, and therefore commute. x 0 u k+1 1 x 0 u k+1 1 v k 1 v 1 u 1 u 2 d Suppose then that the second inequality of (7) is not true, that is |v k+1 1 | < |x 0 u 1 | − 1. Then either v 1 and y 1 are not marked, or (if y 1 = ǫ) the words v 1 and v 2 commute. x 0 u 1 v k 1 v 1 v k 1 y 1 v 2 The proof is now complete. We make use of the previous theorem when proving our second main result. 8
4 Characteristic equation The rest of the paper is devoted to the verification of Theorem 2 Let k ≥ 2 be a positive integer. The system of equations S: {x 0 u i 1 x 1u i 2 x 2 · · ·u i m x m = y 0 v i y 1 | i ∈ N} (12) is equivalent to its subsystem T k given by i ∈ {k, k + 1, k + 2}. In this section we explain the method to be used in the proof of the theorem above. The method was first introduced in [7]. Let X = {x 1 , x 2 , . . ., x k } be a set of unknowns, and e = (w 1 , w 2 ) ∈ X ∗ × X ∗ an equation, such that alph(e) = X. Consider a non-erasing morphism ϕ : X ∗ → Σ ∗ solving e, i.e., ϕ(w 1 ) = ϕ(w 2 ), and denote d i = |ϕ(x i )|. Having got such a solution we choose a new alphabet of unknowns H, construct a new equation e = (w 1 , w 2 ) ∈ H ∗ × H ∗ , and define a length-preserving morphism ϕ : H ∗ → Σ ∗ . The set H consists of letters x i,j , for i = 1, 2, . . ., k and j = 1, 2, . . ., d i . Informally, alphabet H is a set of names of all positions in images of ϕ. This naturally induces the morphism ψ : X ∗ → H ∗ defined by ψ(x i ) = x i,1 · · ·x i,di . With help of that morphism, the equation e is given by w i = ψ(w i ), i = 1, 2. The equation e = (w 1 , w 2 ) is called the characteristic equation of e with respect to the morphism ϕ. Clearly the characteristic equation only depends on the values d i , . . .,d k . Finally, the morphism ϕ is defined by ϕ ◦ ψ = ϕ. It should be clear that ϕ is well defined and length-preserving. Indeed, it maps H into Σ, since ϕ(x i,j ) is the jth letter of ϕ(x i ) for each j and i. The definition also immediately implies that ϕ is a solution of e: ϕ(w 1 ) = ϕ ◦ ψ(w 1 ) = ϕ(w 1 ) = ϕ(w 2 ) = ϕ ◦ ψ(w 2 ) = ϕ(w 2 ). 9
Page 1 and 2: On systems of word equations with s
Page 3 and 4: 2 Preliminaries We suppose that the
Page 5 and 6: does not change if we substitute x
Page 7: comparable, and since primitive roo
Page 11 and 12: contains one occurrence of p. There
Page 13 and 14: Note that the previous argument wou
Page 15: [6] T. Harju, D. Nowotka, On the Eq

On systems of word equations with simple loop sets

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