A characterization of laterally commutative heaps (Q1109059)

Let (\ ): \(Q\times Q\times Q\to Q\) be a ternary operation on a set Q. The author studies the following identities: (i) \(((abc)de)=(a(bcd)e)=(ab(cde))\), i.e. associativity; (ii) \(((abc)de)=(ab(cde))\), i.e. lateral associativity; (iii) \((abc)=(cba)\), i.e. lateral commutativity; (iv) \(((abc)(def)(ghi))=((adg)(beh)(cfi))\), i.e. mediality, and (v) \((abb)=a\), \((bba)=a\), i.e. (\ ) is a Mal'cev function. Theorems: (1) A Mal'cev function is medial iff it is associative and laterally commutative. (2) Every associative and laterally commutative ternary operation is medial. (3) A Mal'cev function is medial iff it is laterally associative and laterally commutative.