Admissible Semimorphisms of icl -Groupoids

If M is an algebra in a semidegenerate congruence-modular variety V , then the set C o n ( M ) of congruences of M is an integral complete l -groupoid (= i c l -groupoid). For any morphism f : M → N of V , consider the map f • : C o n ( M ) → C o n ( N ) , where, for each congruence ε of M , f • ( ε ) is the congruence of N generated by f ( ε ) . Then, f • is a semimorphism of i c l -groupoids, i.e., it preserves the arbitrary joins and the top congruences. The neo-commutative i c l -groupoids were introduced recently by the author as an abstraction of the lattices of congruences of Kaplansky neo-commutative rings. In this paper, we define the admissible semimorphisms of i c l -groupoids. The basic construction of the paper is a covariant functor defined by the following: ( 1 ) to each semiprime and neo-commutative i c l -groupoid A , we assign a coherent frame R ( A ) of radical elements of A ; and ( 2 ) to an admissible semimorphism of i c l -groupoids u : A → B , we assign a coherent frame morphism u ρ : R ( A ) → R ( B ) . By means of this functor, we transfer a significant amount of results from coherent frames and coherent frame morphisms to the neo-commutative i c l -groupoids and their admissible semimorphisms. We study the m -prime spectra of neo-commutative i c l -groupoids and the going-down property of admissible semimorphisms. Using some transfer properties, we characterize some classes of admissible semimorphisms of i c l -groupoids: Baer and weak-Baer semimorphisms, quasi r -semimorphisms, quasi r * -semimorphisms, quasi rigid semimorphisms, etc.