Brandt Semigroup
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Brandt semigroups are completely 0-simple
inverse semigroup In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigr ...
s. In other words, they are
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
and I, J be
non-empty In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whil ...
sets. Define a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
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), where the matrix P is
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek Î ...
with only the
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
''e'' of the group ''G'' in its diagonal.


Remarks

1) The
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s 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'' â‰  0 then there are unique ''x'', ''y'', ''z'' for which ''xa'' = ''a'', ''ay'' = ''a'', ''za'' = ''y''. ::For all idempotents ''e'' and ''f'' nonzero, ''eSf'' â‰  0


See also

* Special classes of semigroups


References

*. Semigroup theory {{abstract-algebra-stub