On Non-denumerable Graphs
نویسندگان
چکیده
PROOF. We shall first prove that every complete graph of power t$i can be split up into the countable sum of trees. Let G be a complete graph of cardinal number ML Let {xa}, an. I t is clear that G = U*=3]Gn and that for each Gn, for every j8 i, there exists one and only one a such that (xa, ^ ) GGM and ce, is a well ordered set of cardinal number tn. We shall first decompose each Tn into four parts Tn,i, i = l, 2, 3, 4, such that Tn,i and rw,2 satisfy the condition: (1) Any two consecutive segments of the graphs Tn,i and Tn,2 are of the form: (xa, xp), (xa, xy), a< /3 , a4i satisfy: (2) Any two consecutive segments of the graphs are of the form: (Xp, Xa), (Xy, Xa)9 j8 < « , J < « , JST^Y-
منابع مشابه
Distinguishability of Infinite Groups and Graphs
The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph Γ is said to have connectivity 1 if there exists a vertex α ...
متن کاملNon-Denumerable Infinitary Modal Logic
Segerberg established an analogue of the canonical model theorem in modal logic for infinitary modal logic. However, the logics studied by Segerberg and Goldblatt are based on denumerable sets of pairs 〈Γ, α〉 of sets Γ of well-formed formulae and well-formed formulae α. In this paper I show how a generalisation of the infinite cut-rule used by Segerberg and Goldblatt enables the removal of the ...
متن کاملGeneralizing König's infinity lemma
1 Introduction D. Kόnig's famous lemma on trees has many applications; in graph theory it is used to extend certain results from finite to infinite graphs (see Nash-Williams [7]); in logic it can be used to prove that a denumerable set of propositional formulas is satisfiable if every finite subset is (see, for example, Van Fraassen [9]). This last result, known as the compactness theorem for p...
متن کاملPresentations of Symbolic Dynamical Systems by Directed Labelled Graphs (notes for a ”mini-cours”, Sda2, Paris 4-5 October 2007)
In symbolic dynamics one studies subshifts (X,SX), X a shift-invariant closed subset of the shift space ΣZ and SX the restriction of the shift on ΣZ to X. An introduction to the theory of subshifts is given in [Ki] and in [LM]. See here also [BP]. The topic of the first talk is a general theory that also includes presentations by compact graphs. The topic of the the second talk are synchronizat...
متن کاملThe Principles of Ambient Calculus Revisited
The paper presents a foundational analysis of the class of ambient calculi focussing on the spatial structures of the processes they can describe. We propose sound correct set-theoretical models for different types of ambient calculi (involving recursion, denumerable parallel composition, etc) by using special types of coalgebras labelled flat systems of equations of set theory. These models he...
متن کاملOn the energy of non-commuting graphs
For given non-abelian group G, the non-commuting (NC)-graph $Gamma(G)$ is a graph with the vertex set $G$ $Z(G)$ and two distinct vertices $x, yin V(Gamma)$ are adjacent whenever $xy neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.
متن کامل