# Brandt semigroup

In mathematics, **Brandt semigroups** are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups:

Let *G* be a group and be non-empty sets. Define a matrix of dimension with entries in

Then, it can be shown that every 0-simple semigroup is of the form with the operation .

As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995). Thus, a Brandt semigroup has the form with the operation .

Moreover, the matrix is diagonal with only the identity element e of the group G in its diagonal.

## Remarks[edit]

1) The idempotents have the form (i,e,i) where e is the identity of G

2) There are equivalent way to define the Brandt semigroup. Here is another one:

ac=bc≠0 or ca=cb≠0 ⇒ a=b

ab≠0 and bc≠0 ⇒ abc≠0

If *a* ≠ 0 then there is unique *x*,*y*,*z* for which *xa* = *a*, *ay* = *a*, *za* = *y*.

For all idempotents *e* and *f* nonzero, *eSf* ≠ 0

## See also[edit]

## References[edit]

- Howie, John M. (1995),
*Introduction to semigroup theory*, Oxford: Oxford Science Publication.