On M-Equivalence and Strong M-Equivalence for Parikh Matrices

Abstract

The notion of strong [Formula: see text]-equivalence was introduced as an order-independent alternative to [Formula: see text]-equivalence for Parikh matrices. This paper further studies the notions of [Formula: see text]-equivalence and strong [Formula: see text]-equivalence. Certain structural properties of [Formula: see text]-equivalent ternary words are presented and then employed to (partially) characterize pairs of ternary words that are ME-equivalent (i.e. obtainable from one another by certain elementary transformations). Finally, a sound rewriting system in determining strong [Formula: see text]-equivalence is obtained for the ternary alphabet.

Strong (2 ⋅ t) and Strong (3 ⋅ t) Transformations for Strong M-Equivalence

“…Suppose Σ = {a, b, c} and w, w ′ ∈ Σ * . It was shown in [11] that whenever w undergoes an αβ-transformation, it suffices to note only the changes in the number of occurrences of abc, acb, and bac as subwords in w. This consequently motivated the following definition.…”

“…Remark 2.10. [11] Suppose Σ = {a, b, c} and w, w ′ ∈ Σ * . Then w s ≡ M w ′ if and only if w ≡ M w ′ with respect to each of the ordered alphabets {a < b < c}, {b < a < c}, and {a < c < b}.…”

“…Lemma 3.10. [11] Suppose Σ = {a, b, c} and w, w ′ ∈ Σ * . Assume that w = xabybaz and w ′ = xbayabz for some x, y, z ∈ Σ * , then the following statements are true:…”

“…To answer that question, we look into the notion of αβ-transformations and MSAE -equivalence, both introduced in [11]. Definition 5.1.…”

“…Suppose Σ = {a, b, c} and w, w ′ ∈ Σ * . It was shown in [11] that whenever w undergoes an αβ-transformation, it suffices to note only the changes in the number of occurrences of abc, acb, and bac as subwords in w. This consequently motivated the following definition. Theorem 5.4 established MSAE -equivalence as a sound rewriting system in determining when two ternary words are strongly M-equivalent.…”