The ternary operations of groupoids

Abstract

We investigate ternary products of groupoids and prove that there is a one-to-one correspondence between the collection of right modular groupoids with a left identity element l and laterally commutative, l-bi-unital semiheaps. This result is applied to prove that the natural ternary product induced by a right modular groupoid S with left identity is isomorphic to the natural ternary product of a groupoid T if and only if S and T are isomorphic groupoids. Ternary products of other classes of groupoids are also characterised, including the natural and standard ternary products of inverse semigroups. These two classes of ternary products are proved to be varieties of type (3,1,1).