In Mathematical, a element absorbing (or element allowed ) of a Ensemble for a Law of composition interns is an element of this unit which transforms all the other elements into the absorbing element when it is combined with them by this law.

Definition

• has is known as element absorbing on the left if $\backslash \; forall\; X\; \backslash \; in\; E,\; \backslash \; quad\; has\; \backslash \; signal\; X\; =\; a$;
• has is known as element absorbing on the right if $\backslash \; forall\; X\; \backslash \; in\; E,\; \backslash \; quad\; X\; \backslash \; signal\; has\; =\; a$;
• has is known as element absorbing if it is absorbing on the right and on the left.

Properties

• In a given magma, the absorbing element, if there exists, is single. Indeed, if $a\_1$ and $a\_2$ are two elements absorbents, $a\_1\; =\; a\_1\; \backslash \; signal\; a\_2\; =\; a\_2$.
• On the other hand, several elements absorbents on the left or on the right can exist in a given magma, but if there exists more than one on the left absorbing element, there are some no on the right, and reciprocally. Indeed, let us suppose $a\_1$ and $a\_2$ two elements absorbents on the left, and $b$ an on the right absorbing element: $a\_1\; =\; a\_1\; \backslash \; signal\; B\; =\; B\; =\; a\_2\; \backslash \; signal\; B\; =\; a\_2$.
• If a magma has an element absorbing on the left and an on the right absorbing element, these two elements are equal and the magma has a asorbant element. Indeed, if $a\_1$ is absorbing on the left and $a\_2$ absorbing on the right, $a\_1\; =\; a\_1\; \backslash \; signal\; a\_2\; =\; a\_2$.
• the absorbing element of an internal law of composition is idempotent by this law: $a\; \backslash \; signal\; has\; =\; a$.

Examples

• the absorbing element of the Multiplication between real numbers is the Zero: $a\; \backslash \; cdot\; 0\; =\; 0\; \backslash \; cdot\; has\; =\; 0$. In a similar way, the null Vecteur is element absorbing for the vector Product and the Empty set is element absorbing for the intersection of units.
• For any unit E, on the together of the parts P (E), E is element absorbing for the meeting of units.
• only the group having an absorbing element is the commonplace Groupe.
• In a ring (has, +, ×), the neutral element of + is absorbing element of ×.
• By considering the whole of the functions defined on $\backslash \; mathbb\; \{R\}$ in value in $\backslash \; mathbb\; \{R\}$, equipped with the law $f\; \backslash \; signal\; G\; =\; F\; \backslash \; circ\; g$, the elements absorbents on the left are the constant functions. And there does not exist on the right absorbing element.

See too

• neutral Element
• symmetrical Element
• invertible Element