# Commutativity

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## Commutativity

a property of the addition and multiplication of numbers expressed by the identities a + b = b + a and ab = ba. In a more general sense, the operation a * b is termed commutative if a * b = b *a. Addition and multiplication of polynomials, for example, have the property of commutativity; vector multiplication (see VECTOR PRODUCT) is not commutative since [a,b] = — [b,a].

References in periodicals archive ?
Wang, "Noise-Free Symmetric Fully Homomorphic Encryption Based on Non-Commutative Rings," IACR Cryptol.
Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction.
Moreover, authors noticed that the main problem for construction of cryptographic primitives in infinite non-commutative groups is to reliably hide the factors in group word.
They cover both commutative and non-commutative non-Archimedean dynamics.
2009, Super, Quantum and Non-Commutative Species, Afr.
6) Spec o G is a contravariant graded functor from the category of non-commutative Zariski filtered rings with units and strict epimorphisms to the category of topological (graded) spaces and continuous maps.
Non-commutative geometry unites the non-commutative operators of quantum mechanics and the geometrical objects of general relativity.
the ring of polynomials is now a non-commutative ring of twisted polynomials, defined over the differential field of meromorphic functions in system variables,
Lanski proved that if R is a prime ring, L a non-commutative Lie ideal of R and g [not equal to] 0 a derivation of R, such that [g(x), x] [member of] Z(R), for all x [member of] L, then either R is commutative, or char(R) = 2 and R satisfies [s.
Then he moves into much more advanced terrain, dealing with how to set up working Monte Carlo simulations of matrix field theories, which involve finite dimensional matrix regularizations of non-commutative and fuzzy field theories, fuzzy spaces, and matrix geometry.
Mathematically, quantum information theory extends Shannon~s theory of information but differs from it by allowing both for stronger correlations known as entanglement and for non-commutative effects resulting in measurement uncertainty as required by the laws of quantum physics.

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