what are some binary operations that commute?

Question

Two binary operations $(\otimes, \oplus)$ commute if(?): $$ (a \otimes b) \oplus (c \otimes d) = (a \oplus c) \otimes (b \oplus d) $$

Firstly, is this the standard way of defining commutative operatioins?

[edits: operator was changed to operation (thanks to @blue) the example was wrong and removed (thanks to @Andreas) ]

I'd call those binary operations, not operators. – anon Jul 30 '14 at 03:30 — anon, Jul 30 '14 at 03:30

score 5 · Accepted Answer · answered Jul 30 '14 at 03:48

First, I don't think "commutative operations" is standard terminology (for this or for anything else).

Second, your example of min-max, where I assume you intend $\otimes$ to be min and $\oplus$ to be max, doesn't work. Suppose, for example, that $a=d=1$ and $b=c=0$. Then the left side of your equation is 0 while the right side is 1.

Third, there's a beautiful result, the Eckman-Hilton Lemma, saying that if two operations satisfy your definition of commuting and if each of them has an identity element, then the two operations are the same and they are (or rather, it is) commutative (in the usual sense) and associative.

I will update the question to remove the mistake. Thanks. – SKV Jul 30 '14 at 03:52 — SKV, Jul 30 '14 at 03:52

score 0 · Answer 2 · answered Dec 20 '16 at 16:28

Binary linear combinations commute.

In more detail, assume

$$a \otimes b := j_0a+j_1b \qquad a \oplus b := k_0a+k_1 b$$

Then:

$(a \otimes b) \oplus (c \otimes d)$

$= k_0(j_0a+j_1b)+k_1(j_0c+j_1d)$

$= k_0j_0a+k_0j_1b+k_1j_0c+k_1j_1d$

$= k_0j_0a+k_1j_0c+k_0j_1b+k_1j_1d$

$= j_0(k_0a+k_1c)+j_1(k_0b+k_1d)$

$= (a \oplus c) \otimes (b \oplus d)$

This works in any commutative semiring. More generally, if $S$ is a commutative semiring, then the Lawvere theory of $S$-modules is a commutative algebraic theory, which basically means that every pair of $S$-linear combination operations commute.

what are some binary operations that commute?

2 Answers2