Magma (algebra) - Classification By Properties

Classification By Properties

Group-like structures
Totality* Associativity Identity Inverses Commutativity
Magma Yes No No No No
Semigroup Yes Yes No No No
Monoid Yes Yes Yes No No
Group Yes Yes Yes Yes No
Abelian Group Yes Yes Yes Yes Yes
Loop Yes No Yes Yes No
Quasigroup Yes No No Yes No
Groupoid No Yes Yes Yes No
Category No Yes Yes No No
Semicategory No Yes No No No

A magma (S, *) is called

  • unital if it has an identity element,
  • medial if it satisfies the identity xy * uz = xu * yz (i.e. (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,
  • left-cancellative if for all x, y, and z, xy = xz implies y = z
  • right-cancellative if for all x, y, and z, yx = zx implies y = z
  • cancellative if it is both right-cancellative and left-cancellative
  • a semigroup if it satisfies the identity x * yz = xy * z (associativity),
  • a semigroup with left zeros if there are elements x for which the identity x = xy holds,
  • a semigroup with right zeros if there are elements x for which the identity x = yx holds,
  • a semigroup with zero multiplication or a null semigroup if it satisfies the identity xy = uv, for all x,y,u and v
  • 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.

If is instead a partial operation, then S is called a partial magma.

Read more about this topic:  Magma (algebra)

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