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 [math]\displaystyle{ I, J }[/math] be non-empty sets. Define a matrix [math]\displaystyle{ P }[/math] of dimension [math]\displaystyle{ |I|\times |J| }[/math] with entries in [math]\displaystyle{ G^0=G \cup \{0\}. }[/math]
Then, it can be shown that every 0-simple semigroup is of the form [math]\displaystyle{ S=(I\times G^0\times J) }[/math] with the operation [math]\displaystyle{ (i,a,j)*(k,b,n)=(i,a p_{jk} b,n) }[/math].
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 [math]\displaystyle{ S=(I\times G^0\times I) }[/math] with the operation [math]\displaystyle{ (i,a,j)*(k,b,n)=(i,a p_{jk} b,n) }[/math], where the matrix [math]\displaystyle{ P }[/math] 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
References
- Howie, John M. (1995), Introduction to Semigroup Theory, Oxford: Oxford Science Publication.
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