Quasigroup

The definition of a quasigroup Q says that the left and right multiplication operators defined by are bijections from Q to itself. A magma Q is a quasigroup precisely when these operators are bijective. The inverse maps are given in terms of left and right division by In this notation the quasigroup identities are Quasigroups have the cancellation property: if ab = ac, then < ...

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Quasigroup, Quasigroup - Definitions, Quasigroup - Examples, Quasigroup - Properties, Quasigroup - Latin squares, Quasigroup - Inverse properties, Quasigroup - Morphisms, Quasigroup - Homotopy and isotopy, Quasigroup - Generalizations, Quasigroup - External link

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Division is most often shown by placing the dividend over the divisor with a horizontal line between them. For example, a divided by b is written . This can be read out loud as "a divided by b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this: . This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters. A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, an ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

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The derivative of the quotient of two functions is given by the quotient rule: There is no general method to integrate the quotient of two functions. ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

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Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus: All four quantities are real numbers. r and s may not both be 0. Division for complex numbers expressed in polar form is simpler and easier to remember than the definition above: Again all four quantities are real numbers. r may not be 0. ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

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The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p/q and r/s by All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication. ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

Read more here: » Division mathematics: Encyclopedia II - Division mathematics - Division of rational numbers

With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. Division can be calculated with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

Read more here: » Division mathematics: Encyclopedia II - Division mathematics - Computing division

Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches. Say that 26 cannot be divided by 10. Give the answer as a decimal fraction or a mixed number, so or . This is the approach usually taken in mathematics. Give the answer as a quotient and a remainder, so remainder 6. Give the quotient as the answer, ...

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Division mathematics, Division mathematics - Notation, Division mathematics - Computing division, Division mathematics - Division of integers, Division mathematics - Division of rational numbers, Division mathematics - Division of real numbers, Division mathematics - Division of complex numbers, Division mathematics - Division of polynomials, Division mathematics - Division in abstract algebra, Division mathematics - Division and calculus

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In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase. Algebraic structure - In the ...

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• Algebraic structure - In the sense of universal algebra
• Algebraic structure - Allowing axioms other than identities
• Algebraic structure - Allowing additional structure
• Algebraic structure - Categories

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In universal algebra, one studies algebraic structures consisting of a set and a collection of operations defined on the set which are required to satisfy certain identities. Simple structures Set: a set is a degenerate algebraic structure, one that has zero operations defined on it Pointed set: a set S with a distinguished element s of S Unary system: a set S with a unary operation, i.e. a function S → S Pointed unary system: a unary system with a distinguish ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

Read more here: » Algebraic structure: Encyclopedia II - Algebraic structure - In the sense of universal algebra

Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s). In this way, every algebraic structure defines a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a ca ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

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One broadening of the concept of algebraic structure is to study sets with operations that must satisfy axioms other than identities. Integral domain: a ring with 0 ≠ 1 that has no zero divisors other than 0 Division ring: an integral domain with an inverse operation (the inverse operation is not defined on the whole set) Field: a commutative division ring Although these structures undoubtedly have an algebraic flavor, they suffer from defects not found in universal algebra. For example, there doe ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

Read more here: » Algebraic structure: Encyclopedia II - Algebraic structure - Allowing axioms other than identities

An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: Qn → Q, such that the equation f(x1,...,xn) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Multary means n-ary for some nonnegative n. An example of a multary quasigroup is an iterated group operation, y = x1 · x2See also:

Quasigroup, Quasigroup - Definitions, Quasigroup - Examples, Quasigroup - Properties, Quasigroup - Latin squares, Quasigroup - Inverse properties, Quasigroup - Morphisms, Quasigroup - Homotopy and isotopy, Quasigroup - Generalizations, Quasigroup - External link

Read more here: » Quasigroup: Encyclopedia II - Quasigroup - Generalizations

A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist). Quasigroup - Homotopy and isotopy. Let Q and P be quasigroups. A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that for all x, y in Q. A quasigroup homomorphism is just a homo ...

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Quasigroup, Quasigroup - Definitions, Quasigroup - Examples, Quasigroup - Properties, Quasigroup - Latin squares, Quasigroup - Inverse properties, Quasigroup - Morphisms, Quasigroup - Homotopy and isotopy, Quasigroup - Generalizations, Quasigroup - External link

Read more here: » Quasigroup: Encyclopedia II - Quasigroup - Morphisms

Formally, a quasigroup (Q, *) is a set Q with a binary operation * : Q × Q → Q (that is, it is a groupoid or magma), such that for all a and b in Q there are unique elements x and y in Q such that The unique solutions to these equations are often written x = a \ b and y = b / a. The operations \ and / are called left and right division. We shall al ...

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Quasigroup, Quasigroup - Definitions, Quasigroup - Examples, Quasigroup - Properties, Quasigroup - Latin squares, Quasigroup - Inverse properties, Quasigroup - Morphisms, Quasigroup - Homotopy and isotopy, Quasigroup - Generalizations, Quasigroup - External link

Read more here: » Quasigroup: Encyclopedia II - Quasigroup - Definitions