zero elements
Let be a semigroup. An element is called a right zero [resp. left zero] if [resp. ] for all .
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 for the zero element of a semigroup.
Proposition 1.
If a groupoid has a left zero and a right zero , then .
Proof.
. ∎
Proposition 2.
If is a left zero in a semigroup , then so is for every .
Proof.
For any , . As a result, is a left zero of . ∎
Proposition 3.
If is the unique left zero in a semigroup , then it is also the zero element.
Proof.
By assumption and the previous proposition, is a left zero for every . But is the unique left zero in , we must have , which means that 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 |