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Encyclopedia > Cancellation property

In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

An element a in a magma (M,*) has the right cancellation property (or is right-cancellative) if for all b and c in M, b * a = c * a always implies b = c.

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 properties.

For example, every quasigroup, and thus every group, is cancellative. In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

To say that an element a in a magma (M,*) is left-cancellative, is to say that the function g: x a * x is injective, so a set monomorphism but as it is a set endomorphism it is a set section, i.e. there is a set epimorphism f such f( g( x ) ) = f( a * x ) = x for all x, so f is a retraction. (The only injective function which has no inverse goes from the empty set to a non empty set, so it can't be undone). Moreover, we can be "constructive" with f taking the inverse in the range of g and sending the rest precisely to a. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... A retraction is a public statement that confirms that a previously made statement was incorrect, invalid, or morally wrong. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the range of a function is the set of all output values produced by that function. ... Results from FactBites:

 Cancellation property - Wikipedia, the free encyclopedia (279 words) In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. An element a in a magma (M,*) has the right cancellation property (or is right-cancellative) if for all b and c in M, b * a = c * a always implies b = c. 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 properties.
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