  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.
