# regular semigroup

## Primary tabs

Defines:
regular, $\pi$-regular, eventually regular, strongly $\pi$-regular, group-bound, inverse semigroup, Clifford semigroup, orthodox semigroup, completely regular, epigroup, regular element, inverse, relative inverse
Type of Math Object:
Definition
Major Section:
Reference
Parent:

## Mathematics Subject Classification

### last remark

seems like a completely regular semigroup is a group

### Re: last remark

HkBst writes:

> seems like a completely regular semigroup is a group

No.

### Re: last remark

Let G_1 and G_2 be two (distinct) groups. Let S be the set formed as the disjoint union of G_1 and G_2. Add a new element 0.

Leave the product of two elements in G_1 as before, and the same with G_2. Define the product of an element in G_1 with an element in G_2 as 0, and vice versa. Let 0 act as a zero element. (This is called, in semigroup theory, the 0-disjoint union)

Now, {0} is a subgroup of S, as are G_1 and G_2. Every element is in one of these, so S is completely regular. However, S is clearly not a group.