Open AccessOn Filter $(\alpha)$-convergence and Exhaustiveness of Function Nets in Lattice Groups and Applications
Author(s) -
Antonıo Boccuto,
Xenofon Dimitriou
Publication year - 2015
Publication title -
journal of mathematics research
Language(s) - English
Resource type - Journals
eISSN - 1916-9809
pISSN - 1916-9795
DOI - 10.5539/jmr.v7n2p56
Subject(s) - mathematics , pointwise , pointwise convergence , filter (signal processing) , lattice (music) , context (archaeology) , discrete mathematics , pure mathematics , combinatorics , topology (electrical circuits) , mathematical analysis , network topology , computer science , paleontology , physics , acoustics , computer vision , biology , operating system
We consider (strong uniform)continuity of thelimit of a pointwise convergent net of latticegroup-valued functions, (strong weak)ex\-hau\-sti\-ve\-ness and (strong)$(\alpha)$-con\-ver\-gen\-ce with respect to a pairof filters, which in the setting of nets aremore natural than the corresponding notionsformulated with respect to a single filter. Somecomparison results are givenbetween such concepts, inconnection with suitable properties of filters.Moreover, some modes of filter(strong uniform) continuity for lattice group-valuedfunctions are investigated, givingsome characterization.As an application, we getsome Ascoli-type theorem in an abstract setting,extending earlier results to the context of filter$(\alpha)$-con\-ver\-gen\-ce.Furthermore, we pose some open problems.