zero elements
Let S be a semigroup. An element z is called a right zero [resp. left zero] if xz=z [resp. zx=z] for all x∈S.
An element which is both a left and a right zero is called a zero element.
A semigroup may have many left zeros or right zeros, but if it has at least one of each, then they are necessarily equal, giving a unique (two-sided) zero element.
More generally, these definitions and statements are valid for a groupoid.
It is customary to use the symbol θ for the zero element of a semigroup.
Proposition 1.
If a groupoid has a left zero 0L and a right zero 0R, then 0L=0R.
Proof.
0L=0L0R=0R. ∎
Proposition 2.
If 0 is a left zero in a semigroup S, then so is x0 for every x∈S.
Proof.
For any y∈S, (x0)y=x(0y)=x0. As a result, x0 is a left zero of S. ∎
Proposition 3.
If 0 is the unique left zero in a semigroup S, then it is also the zero element.
Proof.
By assumption and the previous proposition
, x0 is a left zero for every x∈S. But 0 is the unique left zero in S, we must have x0=0, which means that 0 is a right zero element, and hence a zero element by the first proposition.
∎
Title | zero elements |
Canonical name | ZeroElements |
Date of creation | 2013-03-22 13:02:19 |
Last modified on | 2013-03-22 13:02:19 |
Owner | mclase (549) |
Last modified by | mclase (549) |
Numerical id | 6 |
Author | mclase (549) |
Entry type | Definition |
Classification | msc 20N02 |
Classification | msc 20M99 |
Related topic | Semigroup |
Related topic | NullSemigroup |
Related topic | AbsorbingElement |
Defines | zero |
Defines | zero element |
Defines | right zero |
Defines | left zero |