On the semigroup of monoid endomorphisms of the semigroup \(\mathscr{C}_{+}(a,b)\)

Abstract

Let \(\mathscr{C}_{+}(a,b)\) be the submonoid of the bicyclic monoid which is studied in [8]. We describe monoid endomorphisms of the semigroup \(\mathscr{C}_{+}(a,b)\) which are generated by the family of all congruences of the bicyclic monoid and all injective monoid endomorphisms of \(\mathscr{C}_{+}(a,b)\).