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Definition: Existence of a Neutral Element

Let \((X,\ast)\) be an algebraic structure. If $X$ has an element $e\in X$ with

• \(e\ast x=x\) for all \(x\in (X,\ast)\), then $e$ is called left-neutral (or left-identity, or left-unit)
• \(x\ast e=x\) for all \(x\in (X,\ast)\), then $e$ is called right-neutral (or right-identity, or right-unit).

If $e$ is both, left-neutral and right-neutral, then it is called neutral (or identity, or unit).

Notes

• If "$\ast$" is an "multiplication" operation, we can also say multiplicative neutral element and denote $e$ by $1$.
• If "$\ast$" is an "addition" operation, we can also say additive neutral element and denote $e$ by $0$.

Mentioned in:

Axioms: 1 2
Chapters: 3 4 5 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Examples: 21 22 23
Explanations: 24 25
Lemmas: 26 27
Motivations: 28
Proofs: 29 30 31 32 33 34 35 36 37 38 39 40
Propositions: 41 42
Solutions: 43 44
Theorems: 45 46

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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001