in this paper, we study an equivalence relation on the set of fuzzysubgroups of an arbitrary group g and give four equivalent conditions each ofwhich characterizes this relation. we demonstrate that with this equivalencerelation each equivalence class constitutes a lattice under the ordering of fuzzy setinclusion. moreover, we study the behavior of these equivalence classes under theaction of a...

The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to derive the formulas for counting the sizes of Markov equivalence classes. We ...

the main goal of the present paper is to extend classical results from the measure theory and dynamical systems to the fuzzy subset setting. in this paper, the notion of  dynamic equivalence relation is introduced and then it is proved that this relation is an equivalence relation. also, a new metric on the collection of all equivalence classes is introduced and it is proved that this metric is...

When learning a directed acyclic graph (DAG) model via observational data, one generally cannot identify the underlying DAG, but can potentially obtain a Markov equivalence class. The size (the number of DAGs) of a Markov equivalence class is crucial to infer causal effects or to learn the exact causal DAG via further interventions. Given a set of Markov equivalence classes, the distribution of...

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals ...

This paper deals with chain graph models under alternative AMP interpretation. A new representative of an AMP Markov equivalence class, called the largest deflagged graph, is proposed. The representative is based on revealed internal structure of the AMP Markov equivalence class. More specifically, the AMP Markov equivalence class decomposes into finer strong equivalence classes and there exist...

In this paper, we study an equivalence relation on the set of fuzzysubgroups of an arbitrary group G and give four equivalent conditions each ofwhich characterizes this relation. We demonstrate that with this equivalencerelation each equivalence class constitutes a lattice under the ordering of fuzzy setinclusion. Moreover, we study the behavior of these equivalence classes under theaction of a...