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Magma (algebra)

    In abstract algebra a '''magma''' (also called a '''groupoid''') is a particularly basic kind of algebraic structure Specifically a magma consists of a set ''M'' equipped with a single binary operation ''M'' × ''M'' → ''M'' A binary operation is closure (binary operation)|closed by definition but no other axioms are imposed on the operation
    The term magma for this kind of structure was introduced by Bourbaki; however the term groupoid is a very common alternative Unfortunately the term groupoid also refers to an entirely different kind of algebraic concept described at groupoid

    Types of magmas

    Magmas are not often studied as such; instead there are several different kinds of magmas depending on what axioms one might require of the operationCommonly studied types of magmas include
    • quasigroups—nonempty magmas where division is always possible;
    • loop—quasigroups with identity elements;
    • semigroups—magmas where the operation is associative;
    • monoids—semigroups with identity elements;
    • group—monoids with inverse elements or equivalently associative quasigroups (which are always loops);
    • abelian groups—groups where the operation is commutative

    Free magma

    A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object) It can be described in terms familiar in computer science as the magma of full binary trees with leaves labelled by elements of X The operation is that of joining trees at the root It therefore has a foundational role in syntax
    See also: free semigroup free group

    More definitions

    A magma (S *) is called
    • medial if it satisfies the identity xy * uz = xu * yz (ie (x * y) * (u * z) = (x * u) * (y * z) for all x y u z in S)
    • left semimedial if it satisfies the identity xx * yz = xy * xz
    • right semimedial if it satisfies the identity yz * xx = yx * zx
    • semimedial if it is both left and right semimedial
    • left distributive if it satisfies the identity x * yz = xy * xz
    • right distributive if it satisfies the identity yz * x = yx * zx
    • autodistributive if it is both left and right distributive
    • commutative if it satisfies the identity xy = yx
    • idempotent if it satisfies the identity xx = x
    • unipotent if it satisfies the identity xx = yy
    • zeropotent if it satisfies the identity xx * y = yy * x = xx
    • alternative if it satisfies the identities xx * y = x * xy and x * yy = xy * y
    • power-associative if the submagma generated by any element is associative
    • a semigroup if it satisfies the identity x * yz = xy * z (associativity)
    • a semigroup with left zeros if it satisfies the identity x = xy
    • a semigroup with right zeros if it satisfies the identity x = yx
    • a semigroup with zero multiplication if it satisfies the identity xy = uv
    • a left unar if it satisfies the identity xy = xz
    • a right unar if it satisfies the identity yx = zx
    • trimedial if any triple of its (not necessarily distinct) elements generates a medial submagma
    • entropic if it is a homomorphic image of a medial cancellation magma

    See also


    External links