On Filter $(\alpha)$-convergence and Exhaustiveness of Function Nets in Lattice Groups and Applications

  •  Antonio Boccuto    
  •  Xenofon Dimitriou    


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.

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