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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}.
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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.
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| 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. |
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. |
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| 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. |
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. |
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| 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. |
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. |
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| A $0$-associative $1$-biops is called a semigroup. |
A $0$-associative $1$-biops is called a semigroup. |
| A semigroup with identity element is called a monoid. |
A semigroup with identity element is called a monoid. |
| A monoid with inverses is called a group. |
A monoid with inverses is called a group. |
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| 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 both $(S, +)$ and $(S, \cdot)$ are monoids, is called a rig. |
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| 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 $(0, 1)$-distributive $2$-biops $(S, +, \cdot)$, such that $(S, +)$ is a group and $(S, \cdot)$ is a monoid, is called a ring. |
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| A rig with $0$-inverses is a ring. |
A rig with $0$-inverses is a ring. |
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| A $0$-associative $2$-biops $(S, \cdot, /)$ with $0$-identity such that for every $\{a, b\} \subset S$ we have |
A $0$-associative $2$-biops $(S, \cdot, /)$ with $0$-identity such that for every $\{a, b\} \subset S$ we have |
| $$b = (b / a) \cdot a = (b \cdot a) / a$$ |
$$b = (b / a) \cdot a = (b \cdot a) / a$$ |
| is called a group. |
is called a group. |
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| A $3$-biops $(S, \cdot, /, \backslash)$ such that for every $\{a, b\} \subset S$ we have |
A $3$-biops $(S, \cdot, /, \backslash)$ such that for every $\{a, b\} \subset S$ we have |
| $$a \backslash (a \cdot b) = a \cdot (a \backslash b) = b = (b / a) \cdot a = (b \cdot a) / a$$ |
$$a \backslash (a \cdot b) = a \cdot (a \backslash b) = b = (b / a) \cdot a = (b \cdot a) / a$$ |
| is called a quasigroup. |
is called a quasigroup. |
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| A quasigroup such that for every $\{a, b\} \subset S$ we have $a / a = b \backslash b$ is called a loop. |
A quasigroup such that for every $\{a, b\} \subset S$ we have $a / a = b \backslash b$ is called a loop. |
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| A $0$-associative loop is a group. |
A $0$-associative loop is a group. |
