Towards an advanced modelling of complex economic phenomena. Pretopological and topological uncertainty research tools (Q653877)

In the first part of this book the authors make brief references to ordinary pretopology to later move on to fuzzification. They begin with a principal example based on very singular financial products which have passed all ``tests'' to the extent that new axioms were added to those of the most general pretopology until Moore's uncertain pretopology was reached. After a brief description of the fundamental concepts of uncertain ordinary pretopology, the conditions necessary for the existence of isotone uncertain pretopologies are established. From here the existing relationships in a system are expressed by using a fuzzy graph. The authors have considered that this was susceptible to treatment until the ``closes'' were to be found on one side and the ``opens'' on another. The authors start the second part of this work expounding the axioms which allow the definition of a topology in the most general way, which later allows an interesting game when passing to uncertainty. The second block of this part is dedicated to uncertain topology. The transformation of deterministic structures into uncertain structures has captured all of the authors attention. They are able to isolate three ways in which to incorporate uncertainty. They have tested two of them in the pretopology of uncertainty, using a fuzzy subset as a descriptor of a physical or mental object and starting from a referential set formed by the referentials of the fuzzy subset in the first form and constructing the referential set with fuzzy subsets in the second form. In the third part of this book the authors propose the connection of some operative instruments with elaborated formal structures. In this way, they have dedicated each of the blocks of which this part consists to two interesting but incomplete theories: the theory of clans and the theory of affinities. They think that the connection of the previously existing with the now achieved can provide unquestionable advantages at the time of their use for economic and management problem solving.