The purpose of these notes is to define an equivalence between the natural homology theories associated to operads and the homology of functors over certain categories of operators (PROPs) related to operads.

The isospin chemical potential region is known as the sign-problem free of quantum chromodynamics (QCD). In this paper, we introduce to three-dimensional three-state Potts model mimic dense QCD; e.g., QCD effective with heavy quarks at finite density. We call it QCD-like model. does not have sign problem, but can expect that shares some properties QCD. Since obtain non-approximated spin configu...