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