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Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures
From: 
Gabriel Dos Reis 
Subject: 
Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures 
Date: 
14 Mar 2006 01:43:13 +0100 
"Bill Page" <address@hidden> writes:
 I agree with Martin. One should interpret:

 if Integer has Monoid(*,1)

 as the question of whether F = (*,1) is a functor from the category
 containing Integer to Monoid, the category of monoids.
100% agreed.
 Axiom/Aldor language constraints require us to write

 Integer has Monoid(Integer,*,1)

 Martin has suggested a method using 'extend' in Aldor to make
 such an assertion by:

 extend Integer: Monoid(Integer,+,1)
all that makes sense, except that  upon reflection this a while 
not everythign should be explicit parameter.
Basically, the idea is this. When have (T, op, e) as a monoid, it is
because the three elements together satisfy some relations. If I may
borrow some loose analogy, it is like with implicit functions. The
above means that there are some redundancy in the triple. Here, e is
uniquely defined by op (maybe nonconstructively, but when it exists
it is unique). Consequently the neutral element is a function of op
and T. So I would write
Inetger has Monoid(Integer, *)
extend Integer: Monoid(Integer, +) with {
neutral == 0;
}
 Gaby
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, (continued)
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/09
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 RE: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/09
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
 RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/10
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures,
Gabriel Dos Reis <=
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
 RE: [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/11
 Re: [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/11
 Re: [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Re: BINGO,Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
 RE: [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/13
 Re: [Axiomdeveloper] Re: BINGO,Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10