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

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

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

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 ways to define the Brandt semigroup.

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