Let <structure> be the name of a certain algebraic structure, e.g. magma, group.
If $A$ is a <structure>, we say $B$ is a sub<structure> of $A$, written $B\leq A$, if
$B$ itself is a <structure>;
$B$ shares the same operation(s) with $A$;
(The underlying sets) $B\subseteq A$.
If $B\subset A$, we say $B$ is a proper sub<structure> of $A$, written $B<A$.
Example.Let $(M,*)$ be a monoid. $(N,*)$ is a submonoid of $M$ if it is a monoid and $N\subseteq M$.
See Also.