Nilsemigroup
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In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent.
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 ak=0.
Finite nilsemigroups
Equivalent definitions exists for finite semigroup. A finite semigroup S is nilpotent if, equivalently:
- x 1 … x n = y 1 … y n {\displaystyle x_{1}\dots x_{n}=y_{1}\dots y_{n}} for each x i , y i ∈ S {\displaystyle x_{i},y_{i}\in S}, where n {\displaystyle n} 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 I n = [ a , n ] {\displaystyle I_{n}=[a,n]} a bounded interval of positive real numbers. For x, y belonging to I, define x ⋆ n y {\displaystyle x\star _{n}y} as min ( x + y , n ) {\displaystyle \min(x+y,n)}. We now show that ⟨ I , ⋆ n ⟩ {\displaystyle \langle I,\star _{n}\rangle } is a nilsemigroup whose zero is n. For each natural number k, kx is equal to min ( k x , n ) {\displaystyle \min(kx,n)}. For k at least equal to ⌈ n − x x ⌉ {\displaystyle \left\lceil {\frac {n-x}{x}}\right\rceil }, kx equals n. This example generalize for any bounded interval of an Archimedean ordered semigroup.
Properties
A non-trivial nilsemigroup does not contain an identity element. It follows that the only nilpotent monoid is the trivial monoid.
The class of nilsemigroups is:
- closed under taking subsemigroups
- closed under taking quotients
- closed under finite products
- but is not closed under arbitrary direct product. Indeed, take the semigroup S = ∏ n ∈ N ⟨ I n , ⋆ n ⟩ {\displaystyle S=\prod _{n\in \mathbb {N} }\langle I_{n},\star _{n}\rangle }, where ⟨ I n , ⋆ n ⟩ {\displaystyle \langle I_{n},\star _{n}\rangle } is defined as above. The semigroup S is a direct product of nilsemigroups, however its contains no nilpotent element.
It follows that the class of nilsemigroups is not a variety of universal algebra. However, the set of finite nilsemigroups is a variety of finite semigroups. The variety of finite nilsemigroups is defined by the profinite equalities x ω y = x ω = y x ω {\displaystyle x^{\omega }y=x^{\omega }=yx^{\omega }}.