Events

November 1, 2018 at 10:00 pm

# Ring Theory Seminar | Algebraic Structures on Set of All Binary Operations over a Fixed Set, Nov. 9

Isaac Owusu-Mensah

The Ohio University-Ohio State University ring theory seminar series presents Isaac Owusu-Mensah discussing “Algebraic structures on the set of all Binary Operations over a fixed set” on Friday, Nov. 9, at 4:45 p.m. in Cockins Hall 240, OSU-Columbus.

Owusu-Mensah is a graduate student in Mathematics at Ohio University.

Abstract: In recent years, the word magma has been used to designate a pair of the form $(S,\ast)$ where ∗ is a binary operation on the set S. Inspired by that terminology, we use the notation $M(S)$ (the magma of S) to denote the set of all binary operations on the set S (i.e. all magmas with underlying set S.) In [1], distributivity hierarchy graphs of a set are introduced. Given a set S, its hierarchy graph has M(S) as vertices and there is an edge from one operation, $\ast$, to another one, $\circ$, if $\ast$ distributes over $\circ$ . Given ∗∈ M(S), the set $\text{out}(\ast) = \{\circ \in M(S)|\ast \text{distributes over} \circ \}$ is called the outset of $\ast$. We define an operation that make M(S) a monoid in such a way that each outset is a submonoid. This endowment gives us a possibility to compare the various elements of M(S) with respect to the monoid structure of their outsets. Various properties of the operation mentioned above are considered, including multiple additive structures on M(S) that have it as the multiplicative part of a nearring. (This is a report on an ongoing project with Sergio R. Lopez-Permouth and Asiyeh Rafieipour.)

References: \bibitem{LPRH} Sergio Lopez-Permouth and L. H. Rowen, Distributive hierarchies of binary operations. Advances in rings and modules, 225–242, Contemp. Math., 715, Amer. Math. Soc., Providence, RI, 2018.