Let R be a ring. The free algebra on n indeterminates, X1, ..., Xn, is the ring spanned by all linear combinations of products of the variables. This ring is denoted R<X1, ..., Xn>
Unlike in a polynomial ring, the variables do not commute. For example X1X2 does not equal X2X1.
Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. (For a more general coefficient ring, the same construction works if we take the free module on n generators.)
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Algebra is a branch of mathematics of Arabian origin or transmission which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote operations, and letters to represent number and quantity; it also refers to a particular kind of abstract algebra structure, the algebra over a field.
In advanced studies axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural topology compatible with algebraic structure.
Boolean algebra is the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.
Another general algebraic notion which applies to Boolean algebras is the notion of a freealgebra.
Namely, the free BA on κ is the BA of closed-open subsets of the two element discrete space raised to the κ power.
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