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

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:

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:

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 (Z

_{6}, +) and its subsemigroups, H

_{1}= ({2], [4], [0]}, +) and H

_{2}= ({[0], [3]} , +).

Then (H

_{1}H

_{2},+ ) is not a subsemigroup, since [2] + [3] = [5], the equivalence class [5], but [5] is not a member of H

_{1}ÈH

_{2}.

But if H

_{1 }Í

_{ }H

_{2}then H

_{1}È H

_{2}= H

_{2 }, so, H

_{1}È H

_{2}is a subsemigroup of (S , *).

Theorem 2.3.2:

Theorem 2.3.2:

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

Proof:

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

subsemigroup. This is the smallest subsemigroup, which contains T and is denoted by

**.**

the subsemigroup generated by T

the subsemigroup generated by T

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