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. [ABSTRACT FROM AUTHOR]