ℐ-monotonic convergence of sequences of bi-complex numbers

In this paper we investigate the concept of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -monotonic convergence of sequences of bi-complex numbers. We also discuss some partial order relations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>≼</m:mo> <m:msub> <m:mi>i</m:mi> <m:mn>2</m:mn> </m:msub> </m:msub> </m:math> {_{i₂}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>≼</m:mo> <m:msub> <m:mi>i</m:mi> <m:mi>id</m:mi> </m:msub> </m:msub> </m:math> {_{iid}} between two bi-complex numbers and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -convergence, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -monotonic sequences with respect to different partial order relations, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -Cauchy sequences, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -bounded monotonic sequences of bi-complex numbers. The analysis indicates that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -monotonicity with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>≼</m:mo> <m:msub> <m:mi>i</m:mi> <m:mn>2</m:mn> </m:msub> </m:msub> </m:math> {_{i₂}} is not comparable with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℐ</m:mi> </m:math> {I} -monotonicity with respect to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mo>≼</m:mo> <m:msub> <m:mi>i</m:mi> <m:mi>id</m:mi> </m:msub> </m:msub> </m:math> {_{iid}} .