The Models of Granular System and Algebraic Quotient Space in Granular Computing

Granular computing(Gr C) is an emerging computing paradigm, and it is an umbrella term exploring multilevel granularity. we present a generic abstract mathematical model of the granular system. Supposing the inter-granule structure as an algebra, we propose the algebraic quotient space model. In this model, the granulation is based on a congruence relation and all the congruence relations on a granular system form a complete semi-order lattice, which is the theoretical basis for transformation,composition and decomposition among different granularities. The different granulation rules between the topological quotient space model and the algebraic quotient space model lead to the dissimilarity while composing granularities. A real-world case study is presented that demonstrates how the algebraic quotient space model works in the network transmission by error-correcting code. These work shows that the granular system model and the algebraic quotient space model are powerful conceptual modeling and functional specification methodologies for Gr C.