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A zero element (or an absorbing/annihilating element) is an element z such that for all s in S, z • s = s • z = z. This notion can be refined to the notions of left zero , where one requires only that z • s = z , and right zero , where s • z = z .
An absorbing element in a multiplicative semigroup or semiring generalises the property 0 ⋅ x = 0. Examples include: The empty set, which is an absorbing element under Cartesian product of sets, since { } × S = { } The zero function or zero map defined by z(x) = 0 under pointwise multiplication (f ⋅ g)(x) = f(x) ⋅ g(x)
The role of 0 as additive identity generalizes beyond elementary algebra. In abstract algebra, 0 is commonly used to denote a zero element, which is the identity element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). (Such elements may also be called zero elements.)
Absorbing elements are also sometime called annihilating elements or zero elements. A universe set is an absorbing element of binary union . The empty set is an absorbing element of binary intersection and binary Cartesian product , and it is also a left absorbing element of set subtraction :
Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero. Analogous to the above construction, for every semigroup S , one can define S 0 , a semigroup with 0 that embeds S .
-Neighborhoods are absorbing: This condition gives insight as to why every neighborhood of the origin in every topological vector space (TVS) is necessarily absorbing: If is a neighborhood of the origin in a TVS then for every 1-dimensional vector subspace , is a neighborhood of the origin in when is endowed with the subspace topology induced ...
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. [1] [2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings.
Let a nonempty X ⊆ N be given and assume X has no least element. Because 0 is the least element of N, it must be that 0 ∉ X. For any n ∈ N, suppose for every k ≤ n, k ∉ X. Then S(n) ∉ X, for otherwise it would be the least element of X. Thus, by the strong induction principle, for every n ∈ N, n ∉ X.