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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 ${\displaystyle I,J}$ be non-empty sets. Define a matrix ${\displaystyle P}$ of dimension ${\displaystyle |I|\times |J|}$ with entries in ${\displaystyle G^{0}=G\cup \{0\}.}$

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

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

Remarks

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. 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

See also

• Special classes of semigroups

References

• Howie, John M. (1995), Introduction to Semigroup Theory, Oxford: Oxford Science Publication.

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