Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element a in a magma (M, ∗) has the left cancellation property (or is left-cancellative) if for all b and
Biracks and biquandles
In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory
Small Latin squares and quasigroups
Latin squares and quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of so
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that
Quasifield
In mathematics, a quasifield is an algebraic structure where + and are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are
Bol loop
In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in. A loop, L
Commutative magma
In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to
Power associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Moufang loop
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang. Smooth Moufan
Isotopy of loops
In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop. Isotopy for loops and quasigroups was introduced by Albert, based on hi
CH-quasigroup
In mathematics, a CH-quasigroup, introduced by , definition 1.3), is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH" stands for cubic hypersurface.
Heap (mathematics)
In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted that satisfies a modified associativity property: A biunitary element h of a
Lie-admissible algebra
In algebra, a Lie-admissible algebra, introduced by A. Adrian Albert, is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket [a, b] = ab − ba. Examples include associativ
Moufang polygon
In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of .In a book on th
Monster vertex algebra
The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove
Pre-Lie algebra
In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space. The notion of pre-Lie al
Lie conformal algebra
A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category. Lie conformal algebras are very closely rela
Malcev-admissible algebra
In algebra, a Malcev-admissible algebra, introduced by Myung, is a (possibly non-associative) algebra that becomes a Malcev algebra under the bracket [a, b] = ab − ba. Examples include alternative alg
Medial magma
In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) which satisfies the identity , or more simply for all x, y, u and v, using the co
Admissible algebra
In mathematics, an admissible algebra is a (possibly non-associative) commutative algebra whose enveloping Lie algebra of derivations splits into the sum of an even and an odd part. Admissible algebra
Hyperbolic quaternion
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form where the squares of i, j, and k are +1 and distinct elements of
Gyrovector space
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the conc
Noncommutative Jordan algebra
In algebra, a noncommutative Jordan algebra is an algebra, usually over a field of characteristic not 2, such that the four operations of left and right multiplication by x and x2 all commute with eac
Vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications
Associator
In abstract algebra, the term associator is used in different ways as a measure of the non-associativity of an algebraic structure. Associators are commonly studied as triple systems.
Magma (algebra)
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by
Racks and quandles
In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of kno
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operat
Latin square
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. An example of a