Section 6.2
Monoids

The are various constructions of new semi-groups and monoids from known ones. The first we discuss is the direct product.

Definition

Let (M1,*1) and (M2,*2) be two semi-groups. We define a multiplication on M1 × M2, the Cartesian product of M1 and M2, as follows:

(x1,x2) * (y1,y2) = (x1 *1 y1,x2 *2 y2).

The resulting structure is again a semi-group, usually called the direct product, and denoted by M1 × M2.

If M1 and M2 are monoids, then so is M1 × M2. The unit element of the product is (e1,e2) where ei is the unit of Mi.