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 $|I|\backslash times\; |J|$ with entries in $G^0=G\; \backslash cup\; \backslash \{0\backslash \}.$

Then, it can be shown that every 0-simple semigroup is of the form $S\; =\; (I\backslash times\; G^0\backslash times\; J)$ with the operation $(i,a,j)*(k,b,n)\; =\; (i,a\; p\_\{jk\}\; b,n)$.

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 $S\; =\; (I\backslash times\; G^0\backslash times\; I)$ with the operation $(i,a,j)*(k,b,n)=(i,a\; p\_\{jk\}\; b,n)$, where the matrix $P$ is diagonal with only the identity element e of the group G in its diagonal.