# Absorbent element

An **absorbent element** is a special element of an algebraic structure .

## definition

Let it be the support set of an algebraic structure with a two-digit link . An element is called left- *absorbing* (with respect to ) if the following applies to all :

- .

Analogously, an element is called *right-absorbing* (with respect to ) if the following applies to all :

- .

An element that is both left and right absorbing (with respect to ) is called *absorbing* (with respect to ), sometimes also **zero element** (but this is often also the name of the neutral element of an additively notated semigroup!).

## properties

For a two-digit link on a set there is at most one absorbing element , because the following applies to absorbing elements :

- .

A left or right absorbing element is always idempotent :

- .

In a quasi-group (and thus also in a group ) with at least two elements with there is no (left- / right-) absorbing element , because otherwise or at least the two solutions would not be clearly solvable, as required for quasi-groups .

## Examples

A well-known example is zero, which is an absorbing element in *every* ring , including in the ring of whole numbers , with regard to multiplication : every number multiplied by zero results in zero.

In every limited association there is an absorbing element for both connections: For example, in propositional logic, the true statement regarding the connection with “or” is an absorbing element, the wrong statement regarding the connection with “and” is an absorbing element.

## See also

## literature

- U. Hebisch; H. J. Weinert:
*Half Rings - Algebraic Theory and Applications in Computer Science*. Teubner, Stuttgart 1993. ISBN 3-519-02091-2 .