Granular computing: A theory of “divide/granulate and conquer“

In this paper, foundational issues on ldquodivide/granulate and conquerrdquo are raised and formalized. Namely, could two distinct problems be divided/granulated into the same set of subproblems? Or equivalently, could the same set of subproblems have more than one integrated solutions? To answer these questions, the concept of extension functors in homological algebra is imported. Informally speaking, the ldquomain programrdquo (the high level instructions), called quotient structure, may integrate the ldquoreturns of subroutine calls,rdquo called sub-structure to more than one integrated solutions of the original problem. The extension functor keeps such solution count.