Endomorphisms and anti-endomorphisms of some finite groupoids

In this paper, we study anti-endomorphisms of some finite groupoids. Previously, special groupoids $S(k, q)$ of order $k(1+k)$ with a generating set of $k$ elements were introduced. Previously, the element-by-element description of the monoid of all endomorphisms (in particular, automorphisms) of a given groupoid was studied. It was shown that every finite monoid is isomorphically embeddable in the monoid of all endomorphisms of a suitable groupoid $S(k, q)$. In recent article, we give an element-by-element description for the set of all anti-endomorphisms of the groupoid $S(k, q)$. We establish that, depending on the groupoid $S(k, q)$, the set of all its anti-endomorphisms may be closed or not closed under the composition of mappings. For an element-by-element description of anti-endomorphisms, we study the action of an arbitrary anti-endomorphism on generating elements of a groupoid. With this approach, the anti-endomorphism will fall into one of three classes. Anti-endomorphisms from the two classes obtained will be endomorphisms of given groupoid. The remaining class of anti-endomorphisms, depending on the particular groupoid $S(k, q)$, may either consist or not consist of endomorphisms. In this paper, we study endomorphisms of some finite groupoids $G$ whose order satisfies some inequality. We construct some endomorphisms of such groupoids and show that every finite monoid is isomorphically embedded in the monoid of all endomorphisms of a suitable groupoid $G$. To prove this result, we essentially use a generalization of Cayley's theorem to the case of monoids (semigroups with identity).