[E-Lang] Operator expansion names, Part 1: Intro

[Ben Laurie]

Bill Frantz wrote:
> 
> At 11:20 PM -0700 4/9/01, Mark S. Miller wrote:
> >At 04:28 PM Monday 4/9/01, Bill Frantz wrote:
> >>Normal addition obeys the rule: a + b equals b + a.  If the goal is to
> >>encourage people building types to model the mathematical concepts of group
> >>and ring, then we want:
> >>
> >>If a and b are in T, then a + b is always in T
> >>a + 0 equals a
> >>a + b equals b + a
> >>(a + b) + c equals a + (b + c)
> >>
> >>My naive belief is that bags and sets obey these rules.
> >
> >What's it called when we have these properties except for commutativity?  I
> >don't want to specify that "+" is generally commutative.
> >
> >
> >        Cheers,
> >        --MarkM
> 
> Having found my Dictionary of Applied Mathematics, here is what it says:
> 
> Vocabulary:
> 
> For all elements: a, b, c, 0 in T; the following hold
> 
> If a and b are in T, then a + b is always in T  --  closed over +
> a + 0 equals a  -- identity element
> a + b equals b + a  --  commutative
> (a + b) + c equals a + (b + c)  --  associative
> (-a) + a = 0  --  reciprocal
> 
> Group: one operation, it is associative, has an identity element, and a
> reciprocal exists.
> 
> Ring: two operations (+, *), both are commutative, associative.  *
> distributes over +, subtraction is always possible.

which means that + has an identity and reciprocals, of course
(subtraction not being a term you defined :-) If * also has an identity
then it is known as a "ring with a one" (note that this is different
from a field coz it isn't required to have inverses).

> Field: two operations (+, *), both are commutative, associative.  *
> distributes over +, + identity is 0, * identity is 1, i.e. 0 + a = 1a = a,
> - and / are always possible and unique (except for division by 0),
> reciprocals exist for each operation except * for 0.
> 
> Ralph Hartley wrote:
> >Groups (but groups only) are called abelian or
> >non abelian as well, I don't know why.
> 
> In an Abelian group, the operation is commutative.
> 
> Cheers - Bill
> 
> -------------------------------------------------------------------------
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