Good Pair Equation on Partial Words

Let x, y be partial words and let m, n be positive integers such that x^m is compatible with yⁿ, (x^m ↑ yⁿ), with gcd(m,n) = 1. Assume that for all i ∈ H(x) the word

yⁿ_^ ( i )yⁿ_^ ( i + |x| ). . .yⁿ_^ ( i + ( m - 1 )|x| )

is 1-periodic and that for all i ∈ H(y) the word

x^m_^ ( i )x^m_^ ( i + |y| ). . .x^m_^ ( i + ( n - 1 )|y| )

is 1-periodic. A pair of partial words (x, y) which satisfies this property we will refer to as a “good pair”. Then there exists a partial word z such that x is contained in z ^k, (x ⊂ z ^k), and y ⊂ z ^l for some integers k, l.