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An element a in a magma (M, ∗) has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma (M, ∗) has the left cancellation property (or is left-cancellative) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative ...
In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. [1] In intuitive terms, the cancellation property asserts that from an equality of the form a·b = a·c, where · is a binary operation, one can cancel the element a and deduce the equality b = c.
The cancellation property holds in any integral domain: for any a, b, and c in an integral domain, if a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x ↦ ax is injective for any nonzero a in the domain. The cancellation property holds for ideals in any integral domain: if xI = xJ, then either x is zero or I = J.
A cancellative semigroup is one having the cancellation property: [9] a · b = a · c implies b = c and similarly for b · a = c · a. Every group is a cancellative semigroup, and every finite cancellative semigroup is a group. A band is a semigroup whose operation is idempotent. A semilattice is a semigroup whose operation is idempotent and ...
A monoid (M, •) has the cancellation property (or is cancellative) if for all a, b and c in M, the equality a • b = a • c implies b = c, and the equality b • a = c • a implies b = c. A commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck group construction.
Project cancellation, in government and industry; Cancellation (mail), a postal marking applied to a stamp or stationery indicating the item has been used; Cancellation (insurance), the termination of an insurance policy; Flight cancellation and delay, not operating a scheduled flight
An alternative and more succinct proof follows from the cancellation property. This property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one-to-one map. The result follows from the fact that one-to-one maps on finite sets are permutations.
A quasigroup (Q, ∗) is a non-empty set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), obeying the Latin square property. This states that, for each a and b in Q , there exist unique elements x and y in Q such that both a ∗ x = b {\displaystyle a\ast x=b} y ∗ a = b ...