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# regular semigroup

Let $S$ be a semigroup.

$x\in S$ is *regular* if there is a $y\in S$ such that $x=xyx$.

$y\in S$ is an *inverse*
(or a *relative inverse*) for $x$ if $x=xyx$ and $y=yxy$.

# 1 Regular semigroups

$S$ is a *regular semigroup* if all its elements are regular.
The phrase ’von Neumann regular’ is sometimes used, after the definition for rings.

In a regular semigroup, every principal ideal is generated by an idempotent.

Every regular element has at least one inverse. To show this, suppose $a\in S$ is regular, so that $a=aba$ for some $b\in S$. Put $c=bab$. Then

$a=aba=(aba)ba=a(bab)a=aca$ |

and

$c=bab=b(aba)b=(bab)ab=cab=c(aba)b=ca(bab)=cac,$ |

so $c$ is an inverse of $a$.

# 2 Inverse semigroups

$S$ is an *inverse semigroup* if for all $x\in S$ there is a *unique* $y\in S$ such that $x=xyx$ and $y=yxy$.

In an inverse semigroup every principal ideal is generated by a *unique* idempotent.

In an inverse semigroup the set of idempotents is a subsemigroup, in particular a commutative band.

The bicyclic semigroup is an example of an inverse semigroup. The symmetric inverse semigroup (on some set $X$) is another example. Of course, every group is also an inverse semigroup.

# 3 Motivation

Both of these notions generalise the definition of a group. In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents. Apart from inverse semigroups, there are *orthodox semigroups* where the set of idempotents is a subsemigroup, and *Clifford semigroups* where the idempotents are central.

# 4 Additional

$S$ is called *eventually regular* (or *$\pi$-regular*) if a power of every element is regular.

$S$ is called *group-bound* (or *strongly $\pi$-regular*, or an *epigroup*) if a power of every element is in a subgroup of $S$.

$S$ is called *completely regular* if every element is in a subgroup of $S$.

## Mathematics Subject Classification

20M17*no label found*20M18

*no label found*

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## Comments

## 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.