# 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 : ${\ displaystyle A}$ ${\ displaystyle *}$${\ displaystyle o_ {l} \ in A}$${\ displaystyle *}$${\ displaystyle a \ in A}$

${\ displaystyle o_ {l} * a = o_ {l}}$.

Analogously, an element is called right-absorbing (with respect to ) if the following applies to all : ${\ displaystyle o_ {r} \ in A}$ ${\ displaystyle *}$${\ displaystyle a \ in A}$

${\ displaystyle a * o_ {r} = o_ {r}}$.

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!). ${\ displaystyle *}$${\ displaystyle *}$

## properties

For a two-digit link on a set there is at most one absorbing element , because the following applies to absorbing elements : ${\ displaystyle *}$${\ displaystyle A}$${\ displaystyle o \ in A}$${\ displaystyle o, o '\ in A}$

${\ displaystyle o '= o' * o = o}$.

A left or right absorbing element is always idempotent : ${\ displaystyle o \ in A}$

${\ displaystyle o = o * o}$.

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 . ${\ displaystyle (A, *)}$${\ displaystyle a, b \ in A}$${\ displaystyle a \ neq b}$${\ displaystyle o \ in A}$${\ displaystyle o * x = o}$${\ displaystyle x * o = o}$${\ displaystyle a, b}$

## 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. ${\ displaystyle \ mathbb {Z}}$

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.