Statistical convergence and statistical continuity on locally solid Riesz spaces (Q409701)

Let \(L\) be a real vector space and \(\leq\) be a partial order on this space. The authors give the definition of an ordered vector space as follows:NEWLINENEWLINE(i) if \(x,y\in L\) and \(y \leq x\), then \(y+z\leq x+z\) for each \(z\in L\),NEWLINENEWLINE(ii) if \(x,y\in L\) and \(y \leq x\), then \(\lambda y\leq \lambda x\) for each \(\lambda\geq 0.\)NEWLINENEWLINEIn addition, if \(L\) is a lattice with respect to the partial ordering, then \(L\) is said to be a Riesz space (or a vector lattice).NEWLINENEWLINEThen they introduce the concepts of statistical topological convergence of a sequence, statistical \(\tau\)-boundedness, statistical \(\tau\)-Cauchy property, and statistical continuity in a locally solid Riesz space, which was introduced in [\textit{G. T. Roberts}, Proc. Camb. Philos. Soc. 48, 533--546 (1952; Zbl 0047.10503)]. Moreover, the authors give some results concerning the definitions.