567 research outputs found
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.Comment: 10 pages. Revised version. preprin
Compositions of Functions and Permutations Specified by Minimal Reaction Systems
This paper studies mathematical properties of reaction systems that was
introduced by Enrenfeucht and Rozenberg as computational models inspired by
biochemical reaction in the living cells. In particular, we continue the study
on the generative power of functions specified by minimal reaction systems
under composition initiated by Salomaa. Allowing degenerate reaction systems,
functions specified by minimal reaction systems over a quarternary alphabet
that are permutations generate the alternating group on the power set of the
background set.Comment: 10 pages, preprin
Ramsey Algebra 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 developes 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.Comment: 15 pages. Under revie
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 Salomaa's 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.Comment: 15 pages, preprin
Ramsey Algebras
Hindman's theorem says that every finite coloring of the natural numbers has
a monochromatic set of finite sums. Ramsey algebras are structures that satisfy
an analogue of Hindman's Theorem. This paper introduces Ramsey algebras and
presents some elementary results. Furthermore, their connection to Ramsey
spaces will be addressed.Comment: 14 pages. Minor revision of the previous version. Pre-prin
M-Ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited
The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked
numerous new investigations in the theory of formal languages by various
researchers, among whom is Serbanuta. Recently, a decade-old conjecture by
Serbanuta on the M-ambiguity of words was disproved, leading to new
possibilities in the study of such words. In this paper, we investigate how
selective repeated duplications of letters in a word affect the M-ambiguity of
the resulting words. The corresponding M-ambiguity of those words are then
presented in sequences, which we term as M-ambiguity sequences. We show that
nearly all patterns of M-ambiguity sequences are attainable. Finally, by
employing certain algebraic approach and some underlying theory in integer
programming, we show that repeated periodic duplications of letters of the same
type in a word results in an M-ambiguity sequence that is eventually periodic.Comment: 16 pages, submitted for publication consideratio
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.Comment: 20 page
Ramsey Orderly Algebras as a New Approach to Ramsey Algebras
Ramsey algebras are algebras that induce Ramsey spaces, which are
generalizations of the Ellentuck space and Milliken's space. Previous work
suggests a possible local version of Ramsey algebras induced by infinite
sequences. Hence, we introduce a new structure called orderly algebra. Under
our canonical setup, an algebra is Ramsey if and only if every of its induced
orderly algebra is Ramsey. In this paper, we present justifications for this
novel notion as a sound approach for further study on Ramsey algebras.Comment: 13 pages, preprin
Are Ramsey Algebras Essentially Semigroups
It is known that semigroups are Ramsey algebras. This paper is an attempt to
understand the role associativity plays in a binary system being a Ramsey
algebra. Specifically, we show that the nonassociative Moufang loop of
octonions is not a Ramsey algebra.Comment: 9 pages, presented at the 2016 Asian Mathematical Conference at Bali,
Indonesi
Parikh Motivated Study on Repetitions in Words
We introduce the notion of general prints of a word, which is substantialized
by certain canonical decompositions, to study repetition in words. These
associated decompositions, when applied recursively on a word, result in what
we term as core prints of the word. The length of the path to attain a core
print of a general word is scrutinized. This paper also studies the class of
square-free ternary words with respect to the Parikh matrix mapping, which is
an extension of the classical Parikh mapping. It is shown that there are only
finitely many matrix-equivalence classes of ternary words such that all words
in each class are square-free. Finally, we employ square-free morphisms to
generate infinitely many pairs of square-free ternary words that share the same
Parikh matrix.Comment: 15 pages, preprint submitte
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