# Structure and a duality of binary operations on monoids and groups

## TL;DR

This paper introduces a novel perspective on monoids and groups by exploring the structure of binary operations on sets, revealing a duality and partitioning that deepen understanding of these algebraic objects.

## Contribution

It presents a new duality framework for binary operations on sets that form monoids and groups, along with a characterization distinguishing group operations from others.

## Key findings

- Duality between monoid binary operations and their induced subsets
- Partitioning of group binary operations into isomorphic copies
- New characterization criteria for group binary operations

## Abstract

In this paper we introduce novel views of monoids and groups. More specifically, for a given set $S$, let $S^{S\times S}$ be the set of binary operations on $S$. We equip $S^{S\times S}$ with canonical binary operations induced by the elements of $S$. Let $S^{S\times S}_{mn}$ (respectively, $S^{S\times S}_{gr}$) be the set of binary operations that make $S$ monoids (respectively, groups). Then we have the following "duality": for each $z\in S^{S\times S}_{mn}$ a certain subset of $S^{S\times S}$, denoted by $S^*_z$, is a monoid with a canonical binary operation and is isomorphic to $(S,z)$. If $z\in S^{S\times S}_{gr}$, then $S^{S\times S}_{gr}$ can be partitioned into copies of $S^*_z$. We also give a new characterization of group binary operations which distinguishes them from the other binary operations. These results give us new insights into monoids and groups, and will provide new tools and directions in studying these objects.