Admissible Semimorphisms of icl-Groupoids

If M is an algebra in a semidegenerate congruence-modular variety V, then the set Con(M) of congruences of M is an integral complete l-groupoid (= icl-groupoid). For any morphism f:M→N of V, consider the map f•:Con(M)→Con(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 icl-groupoids, i.e., it preserves the arbitrary joins and the top congruences. The neo-commutative icl-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 icl-groupoids. The basic construction of the paper is a covariant functor defined by the following: (1) to each semiprime and neo-commutative icl-groupoid A, we assign a coherent frame R(A) of radical elements of A; and (2) to an admissible semimorphism of icl-groupoids u:A→B, we assign a coherent frame morphism uρ: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 icl-groupoids and their admissible semimorphisms. We study the m-prime spectra of neo-commutative icl-groupoids and the going-down property of admissible semimorphisms. Using some transfer properties, we characterize some classes of admissible semimorphisms of icl-groupoids: Baer and weak-Baer semimorphisms, quasi r-semimorphisms, quasi r*-semimorphisms, quasi rigid semimorphisms, etc.