Definition talk:Euclidean Domain

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Operations

When setting up a definition of an abstract algebraic structure, I like to explicitly define the operations, e.g. $\left({R, +, \circ}\right)$ and then include such statements as $a = q \circ b + r$ because then it forces the mind to think about the relationships between the operations. Using implicit operations can make the mind say "oh yes I know all about plus and times" and then naturally fall into the trap of assuming that the objects in question are "numbers". This is particularly important when the object is an abstract ring, and particularly applies to definitions.

Not many texts are as fussy as this, but those that are tend to be deeper and more thorough (the insanely detailed 1965: Seth Warner: Modern Algebra comes to mind).

Or does it just provide unnecessary clutter? What do you think? --prime mover 00:26, 1 March 2011 (CST)

I agree that it's important to understand the structure that the definition relates to and writing $\circ$ or $*$ forces you to re-think what is meant by juxtaposition of symbols.
But the latter is a brilliant notation because it groups letters (in terms of space on the page) according to the operation which creates a visual distinction between addition and multiplication, whereas using $\circ$ spaces addition and multiplication approximately equally,

$$ xyz + yzx + zyx + yxz $$ $$ x\circ y\circ z + y\circ z\circ x + z\circ y\circ x + y\circ x\circ z $$

I prefer juxtaposition because it honors visually the mathematical differences between addition and multiplication, whereas other binary operations suggest (to me at least) that the two hold equal places. My view is that once one is used to not trusting multiplication to be commutative or invertible the notation-honors-content philosophy is preferable.
I think a good compromise would be $\circ$ in definitions and the shorter/simpler proofs, but anything longer perhaps note the it is not multiplication of real numbers, then juxtapose. --Linus44 05:39, 1 March 2011 (CST)
"Once one is used to not trusting multiplication to be commutative or invertible" ... My philosophy is that anyone coming on a page at random, knowing (potentially) nothing about the subject under discussion, ought to be able to make sense of it by going back through definitions until one finally reaches somewhere understandable. Under that philosophy, nothing can be taken as read, in theory everything needs to be defined.
I can agree that in the depths of a technical proof, the compact nature of the juxtaposition is more immediately useful than the cumbersome nature of the "general operation", but when you're defining something, I think there's an argument for the verbose notation. And anyway, we can always put brackets round the expressions, and/or (preferably "and") remind the reader that multiplication has precedence over addition if it's appropriate (especially as the notation is no more than a convention, when all said and done).
Depends on the proof. The existing ground-level stuff where distributivity, associativity and commutativity are being used to prove further really basic stuff merits the expanded notation, but higer-level stuff may be better with the abbreviated style.
Your call, though - we have an unwritten gentleman's agreement that the person who first puts the definition up gets ultimate call over the notation used. --prime mover 13:48, 1 March 2011 (CST)
I'm fine with circ, especially on definitions. It shall be so. --Linus44 14:00, 1 March 2011 (CST)
 :-) ... incidentally, do we know why it's "Euclidean"? It would be good to be able to link to the specific result / definition in Euclid's work (most of which is still "in progress", a therapeutic little exercise for a wet Saturday morning) which inspired the definition. If you don't know, we can enter a stub (or raise another template for "this page is complete as far as it goes but it welcomes expansion, namely ..." which I have yet to write). --prime mover 14:36, 1 March 2011 (CST)
I don't know for sure, I had assumed that it was because the Euclidean algorithm works iff it is a Euclidean domain; I have a copy of The Elements so I'll hunt through for the relevant result. -- Linus44 15:26, 1 March 2011 (CST)
Done, I added it to the usual Euclidean algorithm, he didn't work with abstract rings (amateur). --Linus44 16:06, 1 March 2011 (CST)