# biops

Let $S$ be a set and $n\in\mathbf{N}$. Set $\mathbf{N}_{n}:=\{i\in\mathbf{N}|i. 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 $(0,1)$-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