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
Famous quotes containing the words absorbing and/or elements:
“An innocent man is a sin before God. Inhuman and therefore untrustworthy. No man should live without absorbing the sins of his kind, the foul air of his innocence, even if it did wilt rows of angel trumpets and cause them to fall from their vines.”
—Toni Morrison (b. 1931)
“In verse one can take any damn constant one likes, one can alliterate, or assone, or rhyme, or quant, or smack, only one MUST leave the other elements irregular.”
—Ezra Pound (1885–1972)