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 $, I, \times , J,$ with entries in $G^0=G \cup \.$

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

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 $S=(I\times G^0\times I)$ with the operation $(i,a,j)*(k,b,n)=(i,a p_ b,n)$.

Moreover, the matrix $P$ 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

*.

Semigroup theory

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