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

[Bill Frantz]

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.

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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