On mathematical foundations for business modeling

Abstract

Presents sketch-answers to the following three questions. (1) What is a business domain, mathematically? (2) What is a business model of business domain, mathematically? (3) What is the mathematical machinery suitable for building and manipulating business models? These questions may be answered as follows. (1) Any given business domain D is (mathematically) a topos, i.e. a particular case of a quite general mathematical structure. Probably, even the following more refined picture is valid. Each specific kind of business B (banking, insurance, telecom industry, etc.) determines its own kind of toposes, Top(B), so that any business domain D in B is a topos of sort Top(B), (2) Normally, toposes D are infinite and, given such a topos (domain) D, the task of business modeling is to find a finite yet complete presentation of D. Syntactically, this presentation is specified by a (generalized) sketch, i.e. a directed graph with diagrams marked by labels taken from a predefined signature corresponding to Top(B). (In fact, setting Top(B) amounts to nothing but setting some signature of a legitimate predicate and operations). (3) So, thinking semantically, business specifications are sketches, whatever visualization superstructures (ER, OMT, UML) are built over them. Then, a natural mathematical apparatus for managing and manipulating business specifications is the machinery of deriving and rewriting sketches. In essence, the latter is nothing but a counterpart of ordinary logical derivation and algebraic term rewriting for the graph-based situation.