In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity
or more simply,
for all x, y, u and v, using the convention that juxtaposition denotes the same operation but has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, interchange, bi-commutative, bisymmetric, surcommutative, entropic, etc. [1]
Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. The "only if" direction is the Eckmann–Hilton argument. Another class of semigroups forming medial magmas are normal bands. [2] Medial magmas need not be associative: for any nontrivial abelian group with operation + and integers m ≠ n, the new binary operation defined by x • y = mx + ny yields a medial magma that in general is neither associative nor commutative.
Using the categorical definition of product, for a magma M, one may define the Cartesian square magma M × M with the operation
The binary operation • of M, considered as a mapping from M × M to M, maps (x, y) to x • y, (u, v) to u • v, and (x • u, y • v) to (x • u) • (y • v) . Hence, a magma M is medial if and only if its binary operation is a magma homomorphism from M × M to M. This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a Cartesian product. (See the discussion in auto magma object.)
If f and g are endomorphisms of a medial magma, then the mapping f • g defined by pointwise multiplication
is itself an endomorphism. It follows that the set End(M) of all endomorphisms of a medial magma M is itself a medial magma.
The Bruck–Murdoch–Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group A and two commuting automorphisms φ and ψ of A, define an operation • on A by
where c some fixed element of A. It is not hard to prove that A forms a medial quasigroup under this operation. The Bruck–Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way. [3] In particular, every medial quasigroup is isotopic to an abelian group.
The result was obtained independently in 1941 by Murdoch and Toyoda. [4] [5] It was then rediscovered by Bruck in 1944. [6]
The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra [7] if every two operations satisfy a generalization of the medial identity. Let f and g be operations of arity m and n, respectively. Then f and g are required to satisfy
A particularly natural example of a nonassociative medial magma is given by collinear points on elliptic curves. The operation x • y = −(x + y) for points on the curve, corresponding to drawing a line between x and y and defining x • y as the third intersection point of the line with the elliptic curve, is a (commutative) medial magma which is isotopic to the operation of elliptic curve addition.
Unlike elliptic curve addition, x • y is independent of the choice of a neutral element on the curve, and further satisfies the identities x • (x • y) = y. This property is commonly used in purely geometric proofs that elliptic curve addition is associative.
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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 the associative and identity element properties are optional.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.
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 definition. No other properties are imposed.
In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.
In mathematics, and more specifically in abstract algebra, a *-algebra is a mathematical structure consisting of two involutive ringsR and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution.
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices.
In mathematics, a near-ring is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
In mathematics, a band is a semigroup in which every element is idempotent. Bands were first studied and named by A. H. Clifford.
In mathematics, and in particular universal algebra, the concept of an n-ary group is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an n-ary operation is meant any map f: Gn → G from the n-th Cartesian power of G to G. The axioms for an n-ary group are defined in such a way that they reduce to those of a group in the case n = 2. The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte; the first systematic account of polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the Transactions of the American Mathematical Society.
Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative. One of the earliest applications of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols. Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups, and matrix groups have been identified as potential candidates for cryptographic applications. In contrast to non-commutative cryptography, the currently widely used public-key cryptosystems like RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography are based on number theory and hence depend on commutative algebraic structures.
