Let be a set and . Set . If there exists a map where is a binary operation, then I shall say that is an -biops. In other words, an -biops is an algebraic system with binary operations defined on it, and the operations are labelled .
Let be an -biops. If has the property , then I shall say that is a -biops.
For example if is an -biops and is -commutative, -associative, -alternative or -distributive, then I shall say that is a -commutative -biops, -associative -biops, -alternative -biops or -distributive -biops respectively.
If an -biops is - for each then I shall say that is a -biops.
A -distributive -biops , such that both and are monoids, is called a rig.
A -distributive -biops , such that is a group and is a monoid, is called a ring.
A rig with -inverses is a ring.
A -associative -biops with -identity such that for every we have
is called a group.
A -biops such that for every we have
is called a quasigroup.
A quasigroup such that for every we have is called a loop.
A -associative loop is a group.
|Date of creation||2013-03-22 14:44:49|
|Last modified on||2013-03-22 14:44:49|
|Last modified by||HkBst (6197)|