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I would like to hear some intuition about difference between subtraction and division. For binary subtraction operator the standard development is introduction of unary operation of taking negative number. For division however it depends on the domain. For "nice" domains (such as rational and complex numbers) we have unary multiplicative inverse, while for integers the division is adjoint operation introduced via Galois connection, e.g.,

$$ 2 \cdot 5 \leq 11 \Leftrightarrow 5 \leq 11/2 $$

Can binary subtraction be defined via Galois connection as well?

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$2+5\leq 11 \Leftrightarrow 5\leq 11-2$? – Rasmus Jan 12 '11 at 22:49
Perhaps you have meant 7+5≤11⇔5≤11−7, but what do I do for say x+15≤11⇔x≤11−15 in the domain of positive integers? – Tegiri Nenashi Jan 12 '11 at 23:00
@TegiriNenashi It's hard to pick up on what you mean, both in your OP and your last comment. The relationship between subtraction and the nonnegative integers seems entirely analogous to division and the nonzero integers, except for maybe that division does not play as nicely with the ordering as well as subtraction does on the positive integers. I guess I for one have just never thought of subtraction and division in terms of Galois connections. Do you have any references that explain this in these terms? – rschwieb Dec 6 '12 at 14:22
up vote 3 down vote accepted

A context for your question is that of complete lattices (or, more generally, residuated lattices). Given two complete lattices $L,M$ (a complete lattice is a poset $P$ such that for all $S\subseteq P$ the infimum $\bigwedge S$ exists (and it then follows that joins $\bigvee S$ exist as well)) and an order preserving function $f:L\to M$ it holds that $f$ is a left adjoint (resp. right adjoint) iff $f$ preserves joins (resp. meets).

Now, if $L$ is a quantale (which means it is a complete lattice together with an associative binary operation $+:L\times L\to L$ (let's assume unital and commutative as well) such that $\bigvee S+a=(\bigvee S)+a$ holds for all $a\in L$ and $S\subseteq L$) it follows that the function $\square +a:L\to L$, for any $a\in L$ is order preserving and respects joins. Thus it has a right adjoint which we may call $\square -a:L\to L$, satisfying the Galois connection $x+a\le b$ precisely when $x\le b-a$.

Variations are possible of course by considering a binary operation as above that respects meets instead of joins. These are all examples of residuations in complete lattices. In the particular case where $L=[0,\infty]$ and $+$ is addition one obtains truncated subtraction as the right adjoint. Taking the binary operation to be multiplication will result in the residuation being strongly related to division. Many of the familiar properties of these operations can be proved in the much broader context of complete lattices and residuations in them. Perhaps more interestingly is that in quantales that are quite different than $[0,\infty ]$ one still has relatively well-behaved algebraic operations.

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