In mathematics, and more precisely in
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: ''x''·''y'', or simply ''xy'', ...
theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
.
Definitions
Formally, a semigroup ''S'' is a nilsemigroup if:
*''S'' contains ''0'' and
*for each element ''a''∈''S'', there exists a positive integer ''k'' such that ''a''
''k''=''0''.
Finite nilsemigroups
Equivalent definitions exists for finite semigroup. A finite semigroup ''S'' is nilpotent if, equivalently:
*
for each
, where
is the cardinality of ''S''.
*The zero is the only idempotent of ''S''.
Examples
The trivial semigroup of a single element is trivially a nilsemigroup.
The set of
strictly upper triangular matrix, with matrix multiplication is nilpotent.
Let