## Abstract algebra

Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in category theory by Saunders MacLane it has expanded more recently drawing upon work by Kontsevich on graph homology. ...more on Wikipedia about "Operad theory"

(Order of operations) :::exponents and roots ...more on Wikipedia about "Order of operations"

The ordered exponential is the mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function, defined by means of a function from real numbers to a real or complex associative algebra. In practice the values lie in matrix and operator algebras. ...more on Wikipedia about "Ordered exponential"

In abstract algebra, an ordered group is a group G equipped with a partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if ab then agbg and gagb. ...more on Wikipedia about "Ordered group"

In abstract algebra, an ordered ring is a commutative ring $R$ with a a total order $\leq$ such that ...more on Wikipedia about "Ordered ring"

(Orthogonality) In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek ortho, meaning right, and gonia, meaning angle. Two streets that cross each other at a right angle are orthogonal to one another. ...more on Wikipedia about "Orthogonality"

The concept of pairing treated here occurs in mathematics. ...more on Wikipedia about "Pairing"

In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...more on Wikipedia about "Permutation"

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group. The symmetric group of n elements is denoted by Sn; if M is any finite or infinite set, then the group of all permutations of M is often written as Sym(M). ...more on Wikipedia about "Permutation group"

In mathematics, a pointed set is a set X with a distinguished basepoint x0 in X. Maps of pointed sets (based maps) are functions preserving basepoints, i.e. a map f : XY such that f(x0) = y0. This is usually denoted ...more on Wikipedia about "Pointed set"

A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz' law. More precisely, a Poisson algebra is a vector space over a field K equipped with two bilinear products, $\cdot$ and [,] such that $\cdot$ forms an associative K-algebra and [,], called the Poisson bracket, forms a Lie algebra, and for any three elements x, y and z, [x, yz] = [x, y]z + y[x, z] (i.e. the Poisson bracket acts as a derivation). ...more on Wikipedia about "Poisson algebra"

A Poisson superalgebra A is a Z2- graded algebra with two products, . and [,] (which both share the same grading) such that . turns A into an associative algebra, [,] turns A into a Lie superalgebra and the SuperLeibniz Law stating that for any pure element x, [x,.] is a derivation/ antiderivation. ...more on Wikipedia about "Poisson superalgebra"

In mathematics, a polynomial is an expression in which constants and variables are combined using (only) addition, subtraction, and multiplication. Thus, 7x2+4x−5 is a polynomial; 2/x is not. A polynomial function is a function defined by evaluating a polynomial. Polynomial functions are an important class of smooth functions; smooth meaning that they are infinitely differentiable, i.e., they have derivatives of all finite orders. ...more on Wikipedia about "Polynomial"

In mathematics, given a set S, the power set (or powerset) of S, written $\mathcal\left\{P\right\}\left(S\right)$ or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. ...more on Wikipedia about "Power set"

In mathematics, the predual of an object D is an object P whose dual space is D. ...more on Wikipedia about "Predual"

The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logical conjunction and logical disjunction are distributive over each other. The principle is valid in classical logic, but invalid in quantum logic. ...more on Wikipedia about "Principle of distributivity"

In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). A theory of pseudogroups was developed by Élie Cartan around 1920. ...more on Wikipedia about "Pseudogroup"

Pseudoscalar is a concept that can have several meanings in mathematics and physics. They are all related by the fact that a pseudoscalar describes a quantity which behaves almost as a scalar, that is, it has number-like properties. ...more on Wikipedia about "Pseudoscalar"

In mathematics, a rack or a quandle are sets with a binary operation mimicking the three Reidemeister moves of knot theory diagram manipulation. ...more on Wikipedia about "Quandle (knot theory)"

In mathematics, the Quillen–Suslin theorem is a theorem in abstract algebra about the relationship between free modules and projective modules. Projective modules are modules that are locally free. Not all projective modules are free, but in the mid- 1950s, Jean-Pierre Serre found evidence that a limited converse might hold. He asked the question: ...more on Wikipedia about "Quillen–Suslin theorem"

In abstract algebra, a branch of mathematics, given a module and a submodule, one can construct their quotient module. This construction, to be described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings. ...more on Wikipedia about "Quotient module"

In mathematics, there is a concept of a representation of a Hopf algebra. ...more on Wikipedia about "Representation of a Hopf algebra"

In mathematics, if G is a group and H a subgroup, then for any linear representation ρ of G, we can define the restricted representation ...more on Wikipedia about "Restricted representation"

In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...more on Wikipedia about "Ring theory"

In mathematics, a matrix can be thought of as each row or column being a vector. Hence, a space formed by row vectors or column vectors are said to be a row space or a column space. ...more on Wikipedia about "Row and column spaces" This text is made on http://www.shortopedia.com