ickekommutativ

Ickekommutativ is a term used in abstract algebra to describe a restricted form of commutativity for a binary operation on a set. The idea is that the operation behaves like a commutative operation on certain pairs of elements, while it may fail to be commutative for others. The notion is primarily theoretical or exploratory and is not widely standardized in mainstream algebra.

Formally, let S be a set equipped with an equivalence relation ~. A binary operation *: S × S → S is said to be ickekommutativ with respect to ~ if for all a,b in S, a ~ b implies a * b = b * a.

This makes the operation commutative on all pairs connected by ~, while permitting non-commutativity for pairs in different equivalence classes. In particular, standard commutativity (a * b = b * a for all a,b) is recovered when ~ is universal (i.e., every pair is equivalent) or when * is itself fully commutative.

Example: take S = Z with parity relation a ~ b if a ≡ b (mod 2). Define a ⊗ b = h(a) where h(a) = 0 if a is even and 1 if a is odd. Then for any a,b with same parity, a ⊗ b = h(a) = h(b) = b ⊗ a; hence ickekommutativ holds. If a,b have different parity, a ⊗ b ≠ b ⊗ a, so the operation is not commutative in general.

Notes: The concept encapsulates commutativity on equivalence classes and depends on the chosen ~. Different choices of ~ yield different ickekommutativ structures. The term is not standard and may be encountered mainly in speculative or educational contexts.