Parikh matrices have been widely used and researched as a tool for studying numerical properties of words since their introduction by Mateescu et al. [1]. Related notions like M-equivalence, M-unambiguity, and so on have received much attention in the literature. Informatively, two words are M-equivalent if they have a common Parikh matrix, and “a word is M-unambiguous if and only if its M-equivalence class is a singleton.” The most-studied problem related to these concepts is the injectivity of Parikh matrix mappings, and most of the related results in the literature concern binary words only.

In this paper, the authors introduce an interesting notion called the core of a binary word, and show that the injectivity problem for binary words reduces to the same problem restricted to these binary core words. The authors also introduce the notion of core M-unambiguity and obtain a characterization of the set of binary words that are core M-unambiguous.

Another interesting and slightly more generalized concept called the relativized core of a word is also introduced and studied here. This concept refers to the notion of the core of a word relative to a given word for any alphabet. Furthermore, the authors present “some ... conditions for 1-equivalence for higher alphabets in terms of [this concept].” Notably, follow-up work by the first author of this paper, which extends the concepts presented here nicely, appears in the literature [2].