Example

We write out the direct product of the two monoids on two elements. These are M1 and M2, given by the following tables

M1
*
e
a
e
e
a
a
a
e
  
M2
*
e
a
e
e
a
a
a
a

Their direct product M1 × M2 is the monoid on four elements, given by the multiplication table below.

*
(e,e)
(e,a)
(a,e)
(a,a)
(e,e)
(e,e)
(e,a)
(a,e)
(a,a)
(e,a)
(e,a)
(e,a)
(a,a)
(a,a)
(a,e)
(a,e)
(a,a)
(e,e)
(e,a)
(a,a)
(a,a)
(a,a)
(e,a)
(e,a)