biops

Let $S$ be a set and $n\in 𝐍$. Set . If there exists a map $\cdot :{𝐍}_n\to (S^2\to S):i\mapsto {\cdot }_i$ 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 $0,1,\mathrm{\dots },n-1$.

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 $i\in {𝐍}_n$ 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