Definition talk:Translation Mapping

I've always been careful to avoid using $a - b$ in the context of a general abelian group $(G, +)$ because of the tendency to drive the reader down the mental ruts of thinking of the operation $+$ as "just being ordinary arithmetic addition".

There's a tendency I've noticed in books on group theory etc. that the "additive notation" is introduced on a general abelian group with woolly words like "if the operation is (like) addition, then $+$ is (or can be) used" without specifying the fact that addition is a specifically defined operation on a set which is (or extends) the Definition:Minimal Infinite Successor Set.

Admittedly in those group theory books there is no intention to go down that route (they have other fish to fry) and they tend to take the various number structures for granted - but as we have the full analysis of this area at our fingertips, it may be better to keep the language as general as possible and use something like:

"Then the translation by $g$ is the mapping $\tau_g: G \to G$ defined by:
 * $\forall h \in G: \tau_g \left({h}\right) = h + \left({- g}\right)$

where $-g$ is the inverse of $g$."

Alternatively (and I can't remember how far down the route of defining subtraction I've got), we might want to link to a general "subtraction" definition as this crops up repeatedly.

I may be over-thinking this, and you may already have this in mind, but since the issue came up I felt the need to mention it. --prime mover 03:55, 31 March 2012 (EDT)


 * You are right; I will get to it as I am just finishing intensive labour restructuring some definitions. --Lord_Farin 04:01, 31 March 2012 (EDT)
 * Done. --Lord_Farin 04:40, 31 March 2012 (EDT)