There are lots of operations that are not commutative.

I’m looking for striking counter-examples of operations that are not

associative.Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of genuinely lesser importance?

Which role do algebraic structures with non-associative operations play?

There’s a big gap between commutative and non-commuative algebraic structures (e.g. Abelian vs. non-Abelian groups or categories). Both kinds of algebraic structures are of equal importance. Does the same hold for assosiative vs. non-associative algebraic structures?

**Answer**

Subtraction:

(1−2)−3=−4

1−(2−3)=2

**Attribution***Source : Link , Question Author : Hans-Peter Stricker , Answer Author : Martin Argerami*