Subsemigroups and Submonoids

Studying the substructure always helps to understand the whole algebraic structure in depth and detail. Here in this section we
discuss about some basic facts on the subsemigroups and submonoids.

Let (S , *) be a semigroup and T Í S. If the set T is closed under the operation *, then (T , *) is said to be a
subsemigroup of (S , *).

Similarly, let (M , * , e) be a monoid and T Í M. If T is closed under the operation * and e Î T, then (T , * , e) is said
to be a submonoid of (M , * , e).

For example, (N , ·) is a subsemigroup of (Z , ·), the semigroup (E , +) is a subsemigroup of the semigroup of (N, +), where
E is the set of all even positive integers. The semigroup (N , +) is not a submonoid of the monoid (Z , +).

Theorem 2.3.1:

Any nonempty intersection of subsemigroups S_i, of a semigroup (S , *) is again a subsemigroup. In general union of
subsemigroup of (S , *) is not necessarily a subsemigroup.

Proof:

If x and y are in S_i, then x Î S_i and y Î S_i, for all i. Thus, x * y Î S_i , for all i.
Hence, x * y Î S_i. Therefore, S_i is a subsemigroup of (S , *).

Consider the semigroup (Z₆ , +) and its subsemigroups, H₁ = ({2], [4], [0]}, +) and H₂ = ({[0], [3]} , +).
Then (H₁ H₂ ,+ ) is not a subsemigroup, since [2] + [3] = [5], the equivalence class [5], but [5] is not a member of H₁ÈH₂.

But if H₁ Í H₂ then H₁ È H₂ = H₂ , so, H₁ È H₂ is a subsemigroup of (S , *).

Theorem 2.3.2:

For any commutative monoid (M , * , e), the set of idempotent elements of M form a submonoid.

Proof:

Let S be the set of all idempotent elements of (M , * , e). It is clear that e is an idempotent element, since e * e = e.
Thus, S ¹ f .

Let a, b Î S. Then a * a = a and b * b = b.
Now (a * b) * (a * b) = (a * b) * (b * a) (since M is commutative)
= a * (b * b) * a (by associativity)
= a * (b * a) (as b * b = b)
= a * (a * b) (by commutative)
= (a *a) * b (by associativity)
= a * b (as a * a = a).
Therefore S is closed under the operation *.
Hence (S , *) is a submonoid of (M , * , e).

Let (S , *) be a semigroup and but f ¹ T Í S. Then the intersection of all subsemigroup containing T forms a
subsemigroup. This is the smallest subsemigroup, which contains T and is denoted by . Also, we call
the subsemigroup generated by T.