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.