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Statistical convergence and statistical continuity on locally solid Riesz spaces. (English) Zbl 1244.40002
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:
(i) if $$x,y\in L$$ and $$y \leq x$$, then $$y+z\leq x+z$$ for each $$z\in L$$,
(ii) if $$x,y\in L$$ and $$y \leq x$$, then $$\lambda y\leq \lambda x$$ for each $$\lambda\geq 0.$$
In 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).
Then 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 [G. T. Roberts, Proc. Camb. Philos. Soc. 48, 533–546 (1952; Zbl 0047.10503)]. Moreover, the authors give some results concerning the definitions.

##### MSC:
 40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX) 40A35 Ideal and statistical convergence 46A40 Ordered topological linear spaces, vector lattices
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##### References:
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