A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers. While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For example, the octonions are unital, but Lie algebras never are. The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are subalgebras of the full algebra of K-endomorphisms of A as a K-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing A". More generally, some authors consider the concept of a non-associative algebra over a commutative ring R: An R-module equipped with an R-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any R-algebra), then it is naturally a -algebra, so some authors refer to non-associative -algebras as non-associative rings. (Wikipedia).
Intermediate Algebra Lecture 8.2: An Introduction to Non-Linear Functions
https://www.patreon.com/ProfessorLeonard Intermediate Algebra Lecture 8.2: An Introduction to Non-Linear Functions
From playlist Intermediate Algebra (Full Length Videos)
Associative Binary Operations and Examples Video
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Associative Binary Operations and Examples Video. This is video 2 on Binary Operations.
From playlist Abstract Algebra
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From playlist Abstract Algebra
Topics In Noncommutative Algebra and Exponential Growth
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From playlist Algebra
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From playlist Algebra
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From playlist Injective, Surjective, and Bijective Functions
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From playlist Programming
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From playlist Abstract algebra
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From playlist Mathematics
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From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
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From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
A gentle description of a vertex algebra.
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From playlist Vertex Operator Algebras
Sergey Shadrin: Arnold's trinity of algebraic 2d gravitation theories
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From playlist Vertex Operator Algebras
Tony Yue Yu - 4/4 The Frobenius Structure Conjecture for Log Calabi-Yau Varieties
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From playlist Integrable Systems 9th Workshop
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