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

Look up rack or racks in Wiktionary, the free dictionary. Rack or racks may refer to: Amp rack, short for amplifier rack, a piece of furniture in which...

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...

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

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

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

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...

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

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

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

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

known, one may use the Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence...

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

Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or...

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...

Hyperbolic volume Kontsevich invariant Linking number Milnor invariants Racks and quandles and Biquandle Ropelength Seifert surface Self-linking number Signature...

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

M=\bigoplus _{i\in \mathbb {N} }M_{i},} and R i M j ⊆ M i + j {\displaystyle R_{i}M_{j}\subseteq M_{i+j}} for every i and j. Examples: A graded vector space...

order and vice versa. Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any...

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

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

commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization...

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...

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...

(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...

domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. In the 19th century it became a common technique to...

{\displaystyle 1} . This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These...

which is both a unital associative algebra and a counital coassociative coalgebra.: 46  The algebraic and coalgebraic structures are made compatible with...

the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the...