# biops

Let $S$ be a set and $n\in \mathbf{N}$. Set $$. 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 $0,1,\mathrm{\dots},n-1$.

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 $i\in {\mathbf{N}}_{n}$ 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 |