On the monotonicity of functions constructed via ordinal sum construction

The monotonicity of functions defined on the unit interval, constructed via (z)-ordinal sum is discussed. In Part I of this two-part paper we characterize all non-decreasing functions defined on the unit interval which are constructed by a non-trivial ordinal sum of semigroups. We also give necessary and sufficient conditions for a function constructed via ordinal sum to be monotone. In 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 intermediate condition is fulfilled. The case when intermediate condition is not fulfilled is discussed as well. & COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper discusses the monotonicity of functions defined on the unit interval constructed via the (z)-ordinal sum. In Part I, all non-decreasing functions constructed by a non-trivial ordinal sum of semigroups on the unit interval are characterized, and necessary and sufficient conditions for a function constructed via ordinal sum to be monotone are given. In Part II, the structure of a monotone function constructed via the z-ordinal sum with respect to a finite branching set is described, and necessary and sufficient conditions for a function constructed via z-ordinal sum to be monotone when the intermediate condition is fulfilled are provided. The case when the intermediate condition is not fulfilled is also discussed.