Let S be a set and n𝐍. Set 𝐍n:={i𝐍|i<n}. If there exists a map :𝐍n(S2S):ii where i:S2S:(a,b)aib is a binary operationMathworldPlanetmath, then I shall say that (S,) is an n-biops. In other words, an n-biops is an algebraic system with n binary operations defined on it, and the operationsMathworldPlanetmath are labelled 0,1,,n-1.

Let (S,) be an n-biops. If has the property p, then I shall say that (S,) is a p n-biops.

For example if (S,) is an n-biops and is 0-commutativePlanetmathPlanetmathPlanetmath, 0-associative, 0-alternative or (0,1)-distributive, then I shall say that (S,) is a 0-commutative n-biops, 0-associative n-biops, 0-alternative n-biops or (0,1)-distributive n-biops respectively.

If an n-biops B is i-p for each i𝐍n then I shall say that B is a p n-biops.

A 0-associative 1-biops is called a semigroupPlanetmathPlanetmath. A semigroup with identity elementMathworldPlanetmath is called a monoid. A monoid with inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is called a group.

A (0,1)-distributive 2-biops (S,+,), such that both (S,+) and (S,) are monoids, is called a rig.

A (0,1)-distributive 2-biops (S,+,), such that (S,+) is a group and (S,) is a monoid, is called a ring.

A rig with 0-inverses is a ring.

A 0-associative 2-biops (S,,/) with 0-identityPlanetmathPlanetmath such that for every {a,b}S we have


is called a group.

A 3-biops (S,,/,\) such that for every {a,b}S we have


is called a quasigroup.

A quasigroup such that for every {a,b}S we have a/a=b\b is called a loop.

A 0-associative loop is a group.

Title biops
Canonical name Biops
Date of creation 2013-03-22 14:44:49
Last modified on 2013-03-22 14:44:49
Owner HkBst (6197)
Last modified by HkBst (6197)
Numerical id 7
Author HkBst (6197)
Entry type Definition
Classification msc 08A99
Defines semigroup
Defines monoid
Defines group
Defines rig
Defines ring
Defines quasigroup
Defines loop