Normal Omega-Subgroups

Abstract

Subgroups, congruences and normal subgroups are investigated for $\O$-groups. These are lattice-valued algebraic structures, defined on  crisp algebras which are not necessarily groups, and in which the classical equality is replaced by a lattice-valued one. A normal $\O$-subgroup is defined as a particular class in an $\O$-congruence. Our main result is that the quotient groups  over cuts of a normal $\O$-subgroup of an $\O$-group $\overline{\mathcal{G}}$, are classical normal subgroups of the corresponding quotient groups over $\overline{\mathcal{G}}$. We also describe the minimal normal $\O$-subgroup of an $\O$-group, and some other constructions related to $\O$-valued congruences.