Direct Product of Semigroups

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 on S´ T is defined by (s₁ , t₁) (s₂ , t₂) = ( s₁ * s₂ , t₁ D t₂), for any two pairs (s₁ , t₁) and
(s₂, t₂) 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.