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 xS.

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 groupoidPlanetmathPlanetmathPlanetmathPlanetmath.

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.


0L=0L0R=0R. ∎

Proposition 2.

If 0 is a left zero in a semigroup S, then so is x0 for every xS.


For any yS, (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.


By assumptionPlanetmathPlanetmath and the previous propositionPlanetmathPlanetmathPlanetmath, x0 is a left zero for every xS. 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