Title: Monoid Structures on Binary Operations and Distributive Hierarchy Graphs
Speaker: Isaac Owusu Mensah - Ohio University, Athens
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 S. L\'opez-Permouth and A. 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.