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:
“Only those who know the supremacy of the intellectual life ... can understand the grief of one who falls from that serene activity into the absorbing soul-wasting struggle with worldly annoyances.”
—George Eliot [Mary Ann (or Marian)
“Three elements go to make up an idea. The first is its intrinsic quality as a feeling. The second is the energy with which it affects other ideas, an energy which is infinite in the here-and-nowness of immediate sensation, finite and relative in the recency of the past. The third element is the tendency of an idea to bring along other ideas with it.”
—Charles Sanders Peirce (1839–1914)