Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u and v in F , it determines whether there exists a monomorphism of F that sends u to v. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem. Mathematics Subject Classifica...

initiality Lutz Schröder, Horst Herrlich Abstract. We study morphisms that are initial w.r.t. all functors in a given conglomerate. Several results and counterexamples are obtained concerning the relation of such properties to different notions of subobject. E.g., strong monomorphisms are initial w.r.t. all faithful adjoint functors, but not necessarily w.r.t. all faithful monomorphism-preservi...

is a monomorphism for all collections of right R-modules {Ni : i ∈ I} . Restricting the Ni to certain classes of modules one obtains various relativized concepts. For instance, Goodearl investigated this notion relativized to flat modules. The present study provides a uniform background for such relativizations and thus covers and generalizes, among others, a number of known results about such ...

Eu(3+)-doped mixed rare earth orthophosphates (rare earth = La, Y, Gd) have been prepared by hydrothermal technology, whose crystal phase and microstructure both vary with the molar ratio of the mixed rare earth ions. For La(x)Y(1-x)PO(4): Eu(3+), the ion radius distinction between the La(3+) and Y(3+) is so large that only La(0.9)Y(0.1)PO(4): Eu(3+) shows the pure monoclinic phase. For La(x)Gd...

Herein, we have developed a facile and general method for the high-yield fabrication of AgX/Ag(3)PO(4) (X = Cl, Br, I) core-shell heterostructures with an unusual rhombic dodecahedral morphology, which exhibit much higher photocatalytic activities, structural stabilities and photoelectric properties than pure Ag(3)PO(4) crystals in environment and energy applications.