Between associativity and commutativity

For any binary operation, four alternatives exist. It could be (i) both commutative and associative; (ii) neither commutative nor associative; (iii) commutative but not associative; (iv) associative but not commutative. The basic arithmetical operations provide examples for the first two possibilities. This paper presents elementary examples for the last two possibilities. It is claimed that the study of these examples is extremely important in order to understand the logical independence of commutativity and associativity.