In this paper, we make some considerations about admissible orders on the set of closed subintervals of the unit interval I[0,1], i.e. linear orders that refine the product order on intervals. We propose a new way to generate admissible orders on I[0,1] which is more general than those we find in the current literature. Also, we deal with the possibility of an admissible order on I[0,1] to be isomorphic to the usual order on [0,1]. We prove that some orders constructed by our method are not isomorphic to the usual one and we make some considerations about the following question: is there some admissible order on I[0,1] isomorphic to the usual order on [0,1]?

This work was partially supported by the Brazilian funding agency CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), under Proc. No. 307781/2016-0 , and by research projects TIN2016-77356-P (AEI/FEDER,UE) and TIN2016-81731-REDT (AEI/FEDER,UE) of the Spanish Government.