One-object category

Monoid

Consider a two-element set V = {0, 1}, equpped with a binary operaion. multiplication V x V → V, as tabulated below:

0 x 0 = 0

0 x 1 = 0

1 x 0 = 0

1 x 1 = 1

Notice that the number 1 is a two-sided multiplicative identity (satisfying the two identity laws). Also, the binary multiplication (x) satisfies the associative law:

0 x (0 x 0) = 0 x 0 = 0; (0 x 0) x 0 = 0 x 0 = 0

0 x (0 x 1) = 0 x 0 = 0; (0 x 0) x 1 = 0 x 1= 0

0 x (1 x 0) = 0 x 0 = 0; (0 x 1) x 0 = 0 x 0 = 0

0 x (1 x 1) = 0 x 1 = 0; (0 x 1) x 1 = 0 x 1 = 0

1 x (0 x 0) = 1 x 0 = 0; (1 x 0) x 0 = 0 x 0 = 0

1 x (0 x 1) = 1 x 0 = 0; (1 x 0) x 1 = 0 x 1 = 0

1 x (1 x 0) = 1 x 0 = 0; (1 x 1) x 0 = 1 x 0 = 0

1 x (1 x 1) = 1 x 1 = 1; (1 x 1) x 1 = 1 x 1 = 1

Summing it all, the set V = {0, 1}, equipped with the binary multiplication (x), with 1 as its unit satisfying identity and associative laws, is an object of a category: one-object category, or in one word: monoid.