The ternary operations of groupoids

TL;DR

This paper explores the structure of ternary operations in groupoids, establishing correspondences and characterizations that connect different classes of groupoids and their ternary products.

Contribution

It introduces a one-to-one correspondence between right modular groupoids with a left identity and certain semiheaps, and characterizes when ternary products are isomorphic.

Findings

Right modular groupoids with a left identity correspond to certain semiheaps.

Natural ternary products of groupoids are isomorphic iff the groupoids are isomorphic.

Ternary products of inverse semigroups form varieties of specific types.

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).