Wen Chean Teh

Parikh Matrices and Parikh Rewriting Systems

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Parikh rewriting system as a generalization and systematization of Salomaas result. It will be shown that every Parikh rewriting system induces a Thue system without counters that serves as a feasible solution to the injectivity problem.

Core words and Parikh matrices

Parikh matrices have been widely investigated due to their applicability in arithmetizing words by numbers. This paper introduces the core of a binary word, which captures the essential part of a word from the perspective of its Parikh matrix. Additionally, the stronger notion of core M-unambiguity is introduced and the characterization of core M-unambiguous binary words is obtained. Finally, a generalization of the core of a binary word and some of its interesting properties are investigated.

On a conjecture about Parikh matrices

Based on Salomaas characterization of M-equivalence, Atanasiu conjectured that a certain natural generalization of ME-equivalence solves the injectivity problem of Parikh matrices for the ternary alphabet. This paper refutes his conjecture but continues to study the interesting proposed Thue system. Characterization of certain irreducible elementary transformations under this system is obtained. Furthermore, these transformations are further scrutinized in terms of their replaceability by simpler ones.

Parikh Matrices and Strong M-Equivalence

Parikh matrices have been a powerful tool in arithmetizing words by numerical quantities. However, the dependence on the ordering of the alphabet is inherited by Parikh matrices. Strong M-equivalence is proposed as a canonical alternative to M-equivalence to get rid of this undesirable property. Some characterization of strong M-equivalence for a restricted class of words is obtained. Finally, the existential counterpart of strong M-equivalence is introduced as well.

Ramsey Algebras and Formal Orderly Terms

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions of Ramsey algebras by using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.

Separability of M-Equivalent Words by Morphisms

Two words are M-equivalent iff they are indistinguishable by Parikh matrices. Even for the ternary alphabet, an incontestable characterization of the M-equivalence relation is long overdue, ever since the introduction of Parikh matrices by Mateescu et al. in 2001. Recent works by Atanasiu attempted to distinguish M-equivalent words by the Parikh matrices of their images under some morphism. This paper addresses various aspects of this approach. In particular, it is shown that no morphism is capable of completely separating M-equivalent words over a given alphabet. However, if the class of words is restricted in length, then such morphism exists, whose codomain is connected to the notion of t-spectrum.

A New Operator over Parikh Languages

The characterization of M-equivalence for the Parikh matrices is a decade old open problem. This paper studies Parikh matrices and M-equivalence in relation to the s-shuffle operator for the binary alphabet. We also study the distance between images under the s-shuffle operator in a graph associated to the corresponding class of M-equivalent words.

On strongly M-unambiguous prints and Şerbǎnuţǎ's conjecture for Parikh matrices

Abstract In the combinatorial study of words, the Parikh matrix mapping was introduced by Mateescu et al. in 2001 as a natural expansion of the classical Parikh mapping. Solving the general injectivity problem of Parikh matrices remains as one of the most sought after triumph among researchers in this area of study. In this paper, we tackle this problem by extending Şerbǎnuţǎs work regarding prints and M -unambiguity to the context of strong M -equivalence. Consequently, we obtain results on the finiteness of strongly M -unambiguous prints for any finite alphabet. Finally, a related conjecture by Şerbǎnuţǎ is conclusively addressed.

Heterogeneous Ramsey Algebras and Classification of Ramsey Vector Spaces

Carlson introduced the notion of a Ramsey space as a generalization to the Ellentuck space. When a Ramsey space is induced by an algebra, Carlson suggested a study of its purely combinatorial version now called Ramsey algebra. Some basic results for homogeneous algebras have been obtained. In this paper, we introduce the notion of a Ramsey algebra for heterogeneous algebras and derive some basic results. Then, we study the Ramsey-algebraic properties of vector spaces.

Ramsey algebras and the existence of idempotent ultrafilters

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson’s question in some way.