3,102 research outputs found
Every countable model of set theory embeds into its own constructible universe
The main theorem of this article is that every countable model of set theory
M, including every well-founded model, is isomorphic to a submodel of its own
constructible universe. In other words, there is an embedding that
is elementary for quantifier-free assertions. The proof uses universal digraph
combinatorics, including an acyclic version of the countable random digraph,
which I call the countable random Q-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a
set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph,
closely connected with the surreal numbers. The proof shows that contains
a submodel that is a universal acyclic digraph of rank . The method of
proof also establishes that the countable models of set theory are linearly
pre-ordered by embeddability: for any two countable models of set theory, one
of them is isomorphic to a submodel of the other. Indeed, they are
pre-well-ordered by embedability in order-type exactly .
Specifically, the countable well-founded models are ordered by embeddability in
accordance with the heights of their ordinals; every shorter model embeds into
every taller model; every model of set theory is universal for all
countable well-founded binary relations of rank at most ; and every
ill-founded model of set theory is universal for all countable acyclic binary
relations. Finally, strengthening a classical theorem of Ressayre, the same
proof method shows that if is any nonstandard model of PA, then every
countable model of set theory---in particular, every model of ZFC---is
isomorphic to a submodel of the hereditarily finite sets of . Indeed,
is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at
http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe.
(v2 adds a reference and makes minor corrections) (v3 includes further
changes, and removes the previous theorem 15, which was incorrect.
On (4,2)-digraph Containing a Cycle of Length 2
A diregular digraph is a digraph with the in-degree and out-degree of all vertices is constant. The Moore bound for a diregular digraph of degree d and diameter k is M_{d,k}=l+d+d^2+...+d^k. It is well known that diregular digraphs of order M_{d,k}, degree d>l tnd diameter k>l do not exist . A (d,k) -digraph is a diregular digraph of degree d>1, diameter k>1, and number of vertices one less than the Moore bound. For degrees d=2 and 3,it has been shown that for diameter k >= 3 there are no such (d,k)-digraphs. However for diameter 2, it is known that (d,2)-digraphs do exist for any degree d. The line digraph of K_{d+1} is one example of such (42)-digraphs. Furthermore, the recent study showed that there are three non-isomorphic(2,2)-digraphs and exactly one non-isomorphic (3,2)-digraph. In this paper, we shall study (4,2)-digraphs. We show that if (4,2)-digraph G contains a cycle of length 2 then G must be the line
digraph of a complete digraph K_5
Classical properties of algebras using a new graph association
We study the relation between algebraic structures and Graph Theory. We have
defined five different weighted digraphs associated to a finite dimensional
algebra over a field in order to tackle important properties of the associated
algebras, mainly the nilpotency and solvability in the case of Leibniz
algebras
On the digraph of a unitary matrix
Given a matrix M of size n, a digraph D on n vertices is said to be the
digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is
an arc of D. We give a necessary condition, called strong quadrangularity, for
a digraph to be the digraph of a unitary matrix. With the use of such a
condition, we show that a line digraph, LD, is the digraph of a unitary matrix
if and only if D is Eulerian. It follows that, if D is strongly connected and
LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with
some elementary observations. Among the motivations of this paper are coined
quantum random walks, and, more generally, discrete quantum evolution on
digraphs.Comment: 6 page
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