Galois connections between lattices of preradicals induced by ring epimorphisms

We continue the study of Galois connections between the lattices of preradicals of two rings [Formula: see text] and [Formula: see text] induced by an adjoint pair of functors between the categories [Formula: see text]-Mod and [Formula: see text]-Mod. In this paper, we focus on the functor triple induced by any ring homomorphism [Formula: see text], and particularly when it is a ring epimorphism. We give additional results when the epimorphism is flat and when it is projective.

Objective Triangle Functors in Adjoint Pairs

An additive functor [Formula: see text] → [Formula: see text] between additive categories is objective if any morphism f in [Formula: see text] with F(f) = 0 factors through an object K with F(K) = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint (whatever left adjoint or right adjoint) is full or dense. We also give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary adjoint pair (F, G) between categories which are not necessarily additive, we give a sufficient and necessary condition such that F (resp., G) is full or faithful.

FUZZY FILTERS ON THE RESIDUATED LATTICES

In this paper, the lattice operations and the adjoint pair on the fuzzy filters set on residuated lattices are defined, the conclusion that the fuzzy filters lattice defined as such is a distributive residuated lattice is obtained. An order-reversing involution on the fuzzy strong-prime filters sublattice is introduced. It is proved that the fuzzy strong-prime filters sublattice is a quasi-Boolean algebra.

Regularly solvable extensions of non-self-adjoint ordinary differential operators

SynopsisLetL0,M0be closed densely defined linear operators in a Hilbert spaceHwhich form an adjoint pair, i.e.. In this paper, we study closed operatorsSwhich satisfyand are regularly solvable in the sense of Višik. The abstract results obtained are applied to operators generated by second-order linear differential expressions in a weighted spaceL2(a, b; w).