The Kierstead's Conjecture and limitwise monotonic functions

In this paper, we prove Kierstead's conjecture for linear orders whose order types are ∑q∈QF(q), where F is an extended 0′-limitwise monotonic function, i.e. F can take value ζ. Linear orders in our consideration can have finite and infinite blocks simultaneously, and in this sense our result subsumes a recent result of C. Harris, K. Lee and S.B. Cooper, where only those linear orders with finite blocks are considered. Our result also covers one case of R. Downey and M. Moses' work, i.e. ζ⋅η. It covers some instances not being considered in both previous works mentioned above, such as m⋅η+ζ⋅η+n⋅η, for example, where m,n>0.