In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element^{[1]}^{[2]} because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.
Formally, let (S, •) be a set S with a closed binary operation • on it (known as a magma). 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.^{[2]}
Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring. In the case of a semiring with 0, the definition of an absorbing element is sometimes relaxed so that it is not required to absorb 0; otherwise, 0 would be the only absorbing element.^{[3]}
Domain | Operation | Absorber | ||
---|---|---|---|---|
real numbers | ⋅ | multiplication | 0 | |
integers | greatest common divisor | 1 | ||
n-by-n square matrices | matrix multiplication | matrix of all zeroes | ||
extended real numbers | minimum/infimum | −∞ | ||
maximum/supremum | +∞ | |||
sets | ∩ | intersection | ∅ | empty set |
subsets of a set M | ∪ | union | M | |
Boolean logic | ∧ | logical and | ⊥ | falsity |
∨ | logical or | ⊤ | truth |