Absorbing Elements
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 function multiplication, (f × g)(x) = f(x) × g(x), since z × f = z.
Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a field or ring, which is both the additive identity and the multiplicative absorbing element, and whose principal ideal is the smallest ideal.
Read more about this topic: Zero Element
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