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biops (Definition)

Let $ S$ be a set and $ n \in \mathbf{N}$. Set $ \mathbf{N}_n := \{i \in \mathbf{N} \vert 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.

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

$\displaystyle b = (b / a) \cdot a = (b \cdot a) / a$
is called a group.

A $ 3$-biops $ (S, \cdot, /, \backslash)$ such that for every $ \{a, b\} \subset S$ we have

$\displaystyle a \backslash (a \cdot b) = a \cdot (a \backslash b) = b = (b / a) \cdot a = (b \cdot a) / a$
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.



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"biops" is owned by HkBst.
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Also defines:  semigroup, monoid, group, rig, ring, quasigroup, loop
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Cross-references: inverses, identity element, property, binary operation, map
There are 24 references to this entry.

This is version 2 of biops, born on 2004-10-17, modified 2006-09-15.
Object id is 6385, canonical name is Biops.
Accessed 10329 times total.

Classification:
AMS MSC: 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous)
Pending Errata and Addenda
None.
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n biops by remag12 on 2006-11-24 15:17:58
What is the definition of 0-commutative,0-assocaitive,etc.?
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