Structure and Combinatorics on Right Groups

Right groups form an important bridge between group theory and semigroup theory, combining the algebraic symmetry of groups with the one-sided structure of right zero semigroups. This paper develops a complete structural and enumerative theory of right groups. We describe all sub-right groups, Green’s equivalences, Rees’ congruences, and quotients in explicit algebraic terms. Closed formulas and asymptotic estimates are obtained for right groups, sub-right groups, subsemigroups, normal sub-right groups, Rees congruences, and quotient structures, together with a precise comparison between projection and Rees’ quotients. The results unify the structure, combinatorics, and quotient theory of right groups and provide a framework for further investigations into categorical and varietal aspects of right groups.