We consider Lorentz–Karamata spaces, small and grand the so-called L $\mathcal {L}$ , R {R}$ LL {LL}$ LR {LR}$ RL {RL}$ RR {RR}$ spaces. The quasi-norms for a function f in each of these spaces can be defined via nonincreasing rearrangement ∗ $f^*$ or maximal $f^{**}$ . investigate when are equivalent. Most proofs based on Hardy-type inequalities. As an application, we demonstrate how our gener...

Recently, the algebraic theory of MV -algebras is intensively studied. In this paper, we extend the concept of derivation of $MV$-algebras and we give someillustrative examples. Moreover, as a generalization of derivations of $MV$ -algebraswe introduce the notion of $f$-derivations and $(f; g)$-derivations of $MV$-algebras.Also, we investigate some properties of them.

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...