Cyclic subgroupoids of an absolutely free groupoid

Abstract

Subgroupoids of an absolutely free groupoid F = (F,⋅) with a free basis B that are generated by one element (called cyclic subgroupoids of F ) are considered. It is shown that: two cyclic subgroupoids of F have common elements if and only if one of them is contained in the other; F has maximal cyclic subgroupoids and if card(B) ≥ 2 , every cyclic subgroupoid is contained in a maximal one; any two maximal cyclic subgroupoids of F are either disjoint or equal. Also, a characterization of maximal cyclic subgroupoids of F by means of primitive elements in F is given. This statements are also true for an absolutely free groupoid with one-element basis (with modified definition of maximal cyclic subgroupoid).