Isaac Owusu-Mensah
The Undergraduate Mathematics Seminar series features Isaac Owusu-Mensah discussing “Distributive Magmas” on Thursday, Feb. 21, from 5:15 to 6:10 p.m. in Morton 322.
Owusu-Mensah is a graduate student in Mathematics at Ohio University.
Abstract: Binary operations with additional properties have been widely studied: one has semigroups, monoids, groups, etc. In its most general setting, when one has a binary operation over which no additional
properties are assumed, the structure is said to be a magma. We use the notation M(S) (the magma of S) to denote the set of all binary operations on the set S (all magmas on the set S.)
Many interesting algebraic structures deal with a pair of operations, one of which distributes over the other (i.e. an equation similar to a(b+c) = ab + ac holds). Normally, the operations involved satisfy nice properties of their own in addition to the distributivity that relates them. For instance, a ring involves an additive abelian group structure and a monoid multiplicative structure where multiplication distributes over addition. Likewise, a triple (S,o ,*) where (S,o) and (S,*) are magmas is said to be a left (resp. right or two-sided) distributive magma if * left ( resp. right or two-sided) distributes over o. In the absence of other assumptions about * and o, many interesting questions arise.
- How long of a sequence of operations can you have such that o_1 distributes over o_2, o_2 distributes over o_3, etc.?
- Can an operation distribute over itself?
- Can two operations distribute over one another?
These and many other interesting questions will be explored in this presentation, which is part of an ongoing project with Sergio Lopez-Permouth and Asiyeh Rafieipour.
Reference: S. 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
Talks in this seminar series are designed to be accessible to undergraduate mathematics majors, but all are welcome to attend. See more about this seminar series.
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