PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
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.



Anyone with an account can edit this entry. Please help improve it!

"biops" is owned by HkBst.
(view preamble)

View style:

Also defines:  semigroup, monoid, group, rig, ring, quasigroup, loop
Log in to rate this entry.
(view current ratings)

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 9955 times total.

Classification:
AMS MSC: 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
n biops by remag12 on 2006-11-24 15:17:58
What is the definition of 0-commutative,0-assocaitive,etc.?
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)