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 I , J be non-empty sets. Define a matrix P of dimension | I | × | J | with entries in G 0 = G ∪ { 0 } .

Then, it can be shown that every 0-simple semigroup is of the form S = ( I × G 0 × J ) with the operation ( i , a , j ) ∗ ( k , b , n ) = ( i , a p j k b , n ) .

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 S = ( I × G 0 × I ) with the operation ( i , a , j ) ∗ ( k , b , n ) = ( i , a p j k b , n ) .

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