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.
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

1) The idempotents have the form where e is the identity of G.
2) There are equivalent ways 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 are unique x, y, z for which xa = a, ay = a, za = y.
For all idempotents e and f nonzero, eSf ≠ 0