Biops 2004-10-17 12:35:29 2009-01-10 02:26:31 Definition semigroup monoid group rig ring quasigroup loop % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \emph{$n$-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.