On the monotonicity of functions constructed via z-ordinal sum construction

This is the second part of a two-part paper which discusses monotonicity of functions defined on the unit interval, constructed via (z)-ordinal. In Part I we characterized all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups and gave necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In the present Part II we describe the structure of a monotone function defined on the unit interval which is constructed via z-ordinal sum construction with respect to a finite branching set and we give necessary and sufficient conditions for a function constructed via z-ordinal sum to be monotone in the case when the intermediate condition is fulfilled. The case when the intermediate condition is not fulfilled is discussed as well. & COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper is the second part of a two-part series, focusing on the monotonicity of functions defined on the unit interval using (z)-ordinal construction. In Part I, we characterized all non-decreasing functions constructed by a non-trivial ordinal sum of semigroups and provided necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In Part II, we explore the structure of monotone functions constructed via z-ordinal sum with respect to a finite branching set, and we present necessary and sufficient conditions for monotonicity when the intermediate condition is fulfilled. We also discuss the case when the intermediate condition is not fulfilled.