Ramsey Algebras: A Ramseyan Combinatorics For Universal Algebras
Loading...
Date
2018-03
Authors
Teoh, Zu Yao
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
The study of Ramsey algebras is a Ramseyan-type study on algebras. The precise
formulation of a Ramsey algebra is based on the work of Carlson on topological
Ramsey spaces, from which a wide array of classical combinatorial results such as the
Ellentuck theorem and Hindman’s theorem can be derived. After his groundbreaking
work on Ramsey spaces, Carlson suggested that, for spaces that are generated by algebras,
one may pursue a purely combinatorial study of these spaces, where results of
topological nature can be derived from their associated combinatorial results. Such a
direction of study would then be known as Ramsey algebra. The suggestion was first
pursued by Teh in his doctoral work and some basic results concerning homogeneous
algebras were obtained. We begin the thesis with a preliminary, introductory section
required for a cohesive discussion of the subject as well as setting up the required symbols.
We introduce the motivating notions and results for the introduction of Ramsey
algebras into the literature. We then extend the notion of a Ramsey algebra to the more
general setting that encompasses heterogeneous algebras. This is done not only for the
sake of generality, but also heterogeneous structures are ubiquitous in the said work
of Carlson and it is only natural that we consider heterogeneous algebras as well. We
also present some basic results related to heterogeneous algebras before studying some
concrete examples. The concrete examples that we study are the real octonions under
multiplication, vector spaces, and various matrix algebras. We show that the real octonions
do not form a Ramsey algebra under multiplication. Such a study was motivated
by the question as to what role associativity plays in determining semigroups being
Ramsey algebras, whether it is indeed essential for a binary system to be Ramsey.
Description
Keywords
Ramsey algebras is a , study on algebras