Re: [Haskell-cafe] Names for properties of operators

No, they aren't rings, because rings are distributive... 2009/11/8 Thomas Danecker :

1. and 2. are called left- and right-commutative. And I think that 3. and 4. are left- and right-commutative rings (please correct me if I'm wrong here).

Cheers, Thomas

2009/11/7 Neil Brown :

...

Hi,

We have names for properties of operators/functions.  For example, if this holds:

a % b = b % a

for some operator %, we say that % is commutative.  Similarly, if this holds:

(a % b) % c = a % (b % c)

we say that % is associative.  Is there a name for this property, which I'm numbering 1, (where (%) :: a -> b -> b; i.e. the operator is potentially, but not necessarily, asymmetrically typed):

1: a % (b % c) = b % (a % c)

For example, `Set.insert` obeys 1 for any values of a, b and c.  (Any operator that is both associative and commutative automatically satisfies this property, but this property can be satisfied without the operator being either of those.)  Given this property, we could prove useful follow-on results, such as:

foldr (%) x ys = foldr (%) x (reverse ys) foldr (%) x ys = foldl (flip (%)) x ys

The property 1 effectively states that the far-right hand element in a chain of such operators is special, but the ordering of everything to the left of it doesn't matter.

One could conceive of a mirror property (where (%) :: a -> b -> a):

2: (a % b) % c = (a % c) % b

If (%) obeys 1, flip (%) obeys 2 (and vice versa).  I think these properties are useful -- I'd like to know if they have names already to describe them by.  A similar property of two relations (where ((%), (~)) :: (a -> b -> b, c -> b -> b) ) would be:

3: a % (b ~ c) = b ~ (a % c)

with mirror version (and adjusted types):

4: (a % b) ~ c = (a ~ c) % b

Do these have a name?  As an example, `Set.insert` and `Set.union` obey property 3 for all values of a, b and c.

There are also symmetrically-typed examples of these operators, but the Set operations are easy and familiar.

Thanks,

Neil.

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