Let M = (V,*,e) be a monoid.
Definition
A subset W
of V is said to be closed under the multiplication * if,
for all a, b 
W,
the product a * b belongs to W.
A submonoid of M is a subset W of V
closed under multiplication
and containing e.
If W is a submonoid of M, then the restriction of * to W defines a monoid (W,*|W×W,e), which is called the monoid induced on W by M.
The following theorem shows that the intersection of submonoids of a monoid is again a submonoid.
Theorem
H
C
H
If C
is a collection of submonoids of M, then
H
C
H
is also a submonoid of M.
In particular, if W and W' are submonoids of M,
then also W
W' is a submonoid of M.
W' is a submonoid of M.