On I-statistically Cauchy sequences & the class of I- statistically concurrent sequences
Abstract
In this article, we present the ideas of $I$-statistically Cauchy criteria and
$I^{\ast}\!$-statistically Cauchy criteria, which are the generalizations
of $I$-Cauchy and $I^{\ast}\!$-Cauchy criterion, respectively. To grasp the differences,
we compare this $I$-statistically Cauchy criterion with a few other Cauchy criteria.
Also, we investigate
a few characteristics of $I^{\ast}\!$-statistically Cauchy sequences and $I$-statistically
Cauchy sequences and demonstrate their equivalence under the condition that
the ideal $I$ satisfies the property
(AP). Furthermore, a relation is defined on the set $S_X$ of all sequences in a metric
space, which comes out to be
an equivalence relation. Finally, we show that if two sequences belong
to the same equivalence class,
then either both of them are $I$-statistically Cauchy or none of them are.
$I^{\ast}\!$-statistically Cauchy criteria, which are the generalizations
of $I$-Cauchy and $I^{\ast}\!$-Cauchy criterion, respectively. To grasp the differences,
we compare this $I$-statistically Cauchy criterion with a few other Cauchy criteria.
Also, we investigate
a few characteristics of $I^{\ast}\!$-statistically Cauchy sequences and $I$-statistically
Cauchy sequences and demonstrate their equivalence under the condition that
the ideal $I$ satisfies the property
(AP). Furthermore, a relation is defined on the set $S_X$ of all sequences in a metric
space, which comes out to be
an equivalence relation. Finally, we show that if two sequences belong
to the same equivalence class,
then either both of them are $I$-statistically Cauchy or none of them are.
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Subscribers OnlyDOI: https://doi.org/10.2478/tmmp-2024-0013