Granular hierarchical structures of finite naïve subsets and multisets based on free monoids and homomorphisms

Abstract

In this paper, we present granular hierarchical structures in which finite naive subsets and multisets are formulated by means of free monoids and homomorphisms. Our motivation is the observation that we actually write a finite subset as a finite sequence, i.e., string, in the well–known extensive notation. Such correspondence from subsets to strings, however, is not one–to–one because the concept of subsets care neither order nor repetition of their elements. The reason is nothing but a straight consequence from the axiom of extensionality. Then we must represent subsets as equivalence classes of strings with respect to some appropriate equivalence relation based on a homomorphism. Then multisets are also similarly considered as we have granular hierarchal structures of 'strings–multisets–naive subsets'. It should be noted that these formulations only require the numbers how many times each element occurs in every string. Finally the approach is shown to be able to describe every operations of finite subsets and multisets in the level of strings.