From the given two algebraic structures we can always get a bigger algebraic structure by taking the cross product of the two

structures.

Let (S , *) and (T , D ) be two semigroup.

*The*

**direct product**of (S , *) and (T , D ) is the algebraic structure (S´ T,*),*

where the operation

where the operation

*on S´ T is defined by (s*

_{1 }, t_{1})*(s*

(s´

_{2 }, t_{2}) = ( s_{1}* s_{2 }, t_{1}D t_{2}), for any two pairs (s_{1 }, t_{1}) and(s

_{2}, t_{2}) S*T.*

From the definition it follows that (S´ T, ) is a semigroup because the binary operation in S´ T is defined in terms of the

operations * and D and both are associative, so the new operation ‘’ is also associative in S´ T. Thus, (S´ T, ) is a

semigroup. Further, if S and T both are monoids with e and e¢ be their respective identity elements then the element (e, e¢ ) of

S´ T acts as an identity element. Hence (S´ T , ) is a monoid.

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