Equivalence Classes of Maximal Orders

Markov Equivalence Classes for Maximal Ancestral Graphs

Ancestral graphs provide a class of graphs that can encode conditional independence re­ lations that arise in directed acyclic graph (DAG) models with latent and selection vari­ ables, corresponding to marginalization and conditioning. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We introduce a simple representation of a Markov equiv­ ale...

The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Some equivalence classes of operators on B(H)

Maximal Orders in Semigroups1

Inspired by the ring theory concepts of orders and classical rings of quotients, Fountain and Petrich introduced the notion of a completely 0-simple semigroup of quotients in [19]. This was generalised to a much wider class of semigroups by Gould in [20]. The notion extends the well known concept of group of quotients [8]. To give the definition we first have to explain what is meant by a squar...

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassés recent proof of Gallai’s conjecture. We explore this notion further : we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders ar...