Events

October 1, 2019 at 9:39 pm

# Ring Theory Seminar | Monoid Structures on Binary Operations and Distributive Hierarchy Graphs, Oct. 11

The Ohio University-Ohio State University ring theory seminar series presents Isaac Owusu Mensah discussing “Monoid structures on binary operations and Distributive hierarchy graphs” on Friday, Oct. 11, at 4:45 p.m. in Cockins Hall 240, OSU-Columbus.

Mensah is a Teaching Assistant in Mathematics at Ohio University.

#### Abstract

Let \$S\$ be a set and \$M(S)\$ the set of all binary operations on S.  Using the terminology of \cite{LPRH} the (right) distributive hierarchy graph of S is a graph \$H(S)\$ having the elements of \$M(S)\$ as vertices and such that there is an edge from \$\star\$ to  \$\circ\$ if and only if \$\star\$ distributes (on the right) over \$\circ\$.  This graph theoretic visualization lends itself to many natural questions; when the set is finite, combinatorial questions about the distributive hierarchies arise easily.  For instance, one may look for the largest cardinality of a set of vertices \$ X \subset M(S)\$ such that the full subgraph of \$H(S)\$ having \$X\$ as its set of vertices is complete (such a set is called a distributive set of binary operations.)

\cite{PRZY} introduced a monoid structure \$(M(S), \square)\$ on \$M(S)\$ and \cite{MEZ} showed that every group \$(S,\circ)\$ embeds in the monoid \$(M(S), \square)\$; in particular, they showed that the image \$X\$ of \$S\$ in \$M(S)\$ is a right distributive set of binary operations, setting a lower bound of \$n\$ (the cardinality of \$S\$) for the parameter  proposed above.

We investigate a different monoid structure \$(M(S), \triangleleft)\$ on M(S) and consider its units.  We show that among the units of \$(M(S), \triangleleft)\$ is a right distributive set of binary operations which is a group under \$ \triangleleft\$ isomorphic to \$S_n\$, thus improving the lower bound described before from \$n\$ to \$n!\$ .

Other interesting features of the monoid \$(M(S), \triangleleft)\$ will be presented as time allows.

This talk includes results obtained in collaborations with Sergio L’opez-Permouth and Asiyeh Rafieipour.

#### References

\bibitem{LPRH} L\’opez-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.

\bibitem{PRZY} J. H. Przytycki, Distributivity versus associativity in the homology theory of algebraic structures, Demonstratio Math., 44(4), December 2011, 823-869.

\bibitem{MEZ} Mezera, Gregory. Embedding groups into distributive subsets of the monoid of binary operations. Involve 8 (2015), no. 3, 433–437. doi:10.2140/involve.2015.8.433. https://projecteuclid.org/euclid.involve/1511370886