In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams...

Rack of lamb, a cut of meat Rack-rent, a type of property rent Rack, a slang term for a women's breasts Racks and quandles, concepts in abstract algebra...

have certain properties of algebraic and combinatorial interest. They occur in the study of racks and quandles. For any nonnegative integer n, the n-th...

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots...

operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations...

{\displaystyle \mathbb {Z} /6\mathbb {Z} } is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. More generally, for a...

with two binary operations called addition and multiplication, which obey the same basic laws as addition and multiplication of integers, except that multiplication...

groups. A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if: N is a group...

A monoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an identity element, thus obeying all but one...

algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the set of scalars is a field and acts...

sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid...

is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain...

multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The...

integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains. Principal ideal...

field of fractions K and let L be a field extension of K. Then x∈L is integral over A if and only if it is algebraic over K and its minimal polynomial...

special way. Groups with operators were extensively studied by Emmy Noether and her school in the 1920s. She employed the concept in her original formulation...

which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite...

element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set...

of irreducible elements, uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with...

addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental...

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting...

mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied...

theories Operadic algebra Diagrammatic algebra Quantum field theory Racks and quandles Mathematics portal Science portal Technology portal Coherent states...

(with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0. Complements...

algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition...

(1965), p.94. Rutherford (1965), Th.32.1 p.92. Rutherford (1965), p.89. Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd....

satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation:...

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...

equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives...

multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K). The addition and multiplication...