biops (original) (raw)

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-commutative, 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 i∈𝐍n then I shall say that B is a p n-biops.

A 0-associative 1-biops is called a semigroup. A semigroup with identity element is called a monoid. A monoid with inverses 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-identity 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.