biops

Let $S$ be a set and $n\\in\\mathbf{N}$ . Set $\\mathbf{N}_{n}:=\\{i\\in\\mathbf{N}|i<n\\}$ . If there exists a map $\\cdot:\\mathbf{N}_{n}\\to(S^{2}\\to S):i\\mapsto\\cdot_{i}$ where $\\cdot_{i}:S^{2}\\to S:(a,b)\\mapsto a\\cdot_{i}b$ is a binary operation, then I shall say that $(S,\\cdot)$ is an $n$ -biops. In other words, an $n$ -biops is an algebraic system with $n$ binary operations defined on it, and the operations are labelled $0,1,\\ldots,n-1$ .

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

For example if $(S,\\cdot)$ is an $n$ -biops and $\\cdot$ is $0$ -commutative, $0$ -associative, $0$ -alternative or $(0,1)$ -distributive, then I shall say that $(S,\\cdot)$ 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 $i\\in\\mathbf{N}_{n}$ 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 $(S,+,\\cdot)$ , such that both $(S,+)$ and $(S,\\cdot)$ are monoids, is called a rig.

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

A rig with $0$ -inverses is a ring.

A $0$ -associative $2$ -biops $(S,\\cdot,/)$ with $0$ -identity such that for every $\\{a,b\\}\\subset S$ we have

b=(b/a)\\cdot a=(b\\cdot a)/a

is called a group.

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

a\\backslash(a\\cdot b)=a\\cdot(a\\backslash b)=b=(b/a)\\cdot a=(b\\cdot a)/a

is called a quasigroup.

A quasigroup such that for every $\\{a,b\\}\\subset S$ we have $a/a=b\\backslash b$ 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