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.

See also

• Neutral element
• Inverse element

literature

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