An absorbent element is a special element of an algebraic structure .
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!).
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 .
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.
- U. Hebisch; H. J. Weinert: Half Rings - Algebraic Theory and Applications in Computer Science . Teubner, Stuttgart 1993. ISBN 3-519-02091-2 .