Ring Theory Seminar - Isaac Owusu-Mensah

November 6, 2018
Friday, November 9, 2018 - 4:45pm to 5:45pm
Cockins Hall 240
Ring Theory Seminar

Title: Algebraic structures on the set of all Binary Operations over a fixed set

Speaker: Isaac Owusu-Mensah (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 ∗$\in$ 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.)

  1. S. López-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.
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