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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 $\le$ 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 $y\le x$, then $y+z\le x+z$ for each $z\in L$,

(ii) if $x,y\in L$ and $y\le x$, then $\lambda y\le \lambda x$ for each $\lambda \ge 0·$

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 40A35 Ideal and statistical convergence 46A40 Ordered topological linear spaces, vector lattices