Properties [of logical primitives] Page 813 lists theorems satisfied by each function. {0, 1, 6, 7, 8, 9, 14, 15} are commutative (orderless) so that a ∘ b ＝ b ∘ a , while {0, 6, 8, 9, 10, 12, 14, 15} are associative (flat), so that a ∘ (b ∘ c) ＝ (a ∘ b) ∘ c .

But in updating networks a particularly straightforward implementation of one scheme can be obtained if one uses instead a more explicit symbolic representation such as u[1  v[2, 3, 4], 2  v[1, 3, 4], 3  v[1, 2, 4], 4  v[1, 2, 3]] This allows one to capture the basic character of networks by Attributes[u] = {Flat, Orderless}; Attributes[v] = Orderless Updating rules can then be written in terms of ordinary Mathematica patterns.

But while it was immediately clear that most cellular automata do not have the kind of reversible underlying rules assumed in traditional statistical mechanics, it still seemed initially very surprising that their overall behavior could be so elaborate—and so far from the complete orderlessness one might expect on the basis of traditional ideas of entropy maximization.

If f is commutative ( Orderless ) then all that can ever matter to the value of an element is its number of a 's.

And in fact whenever h is commutative ( Orderless ) it turns out that such methods can be used, and substantial speedups obtained.

To get all the familiar properties of additivity one needs an addition operation that is associative ( Flat ) and commutative ( Orderless ), and has an identity element (white or 0 in the cases above)—so that it defines a commutative monoid.