32,042 research outputs found
Sublattices of complete lattices with continuity conditions
Various embedding problems of lattices into complete lattices are solved. We
prove that for any join-semilattice S with the minimal join-cover refinement
property, the ideal lattice IdS of S is both algebraic and dually algebraic.
Furthermore, if there are no infinite D-sequences in J(S), then IdS can be
embedded into a direct product of finite lower bounded lattices. We also find a
system of infinitary identities that characterize sublattices of complete,
lower continuous, and join-semidistributive lattices. These conditions are
satisfied by any (not necessarily finitely generated) lower bounded lattice and
by any locally finite, join-semidistributive lattice. Furthermore, they imply
M. Ern\'e's dual staircase distributivity. On the other hand, we prove that the
subspace lattice of any infinite-dimensional vector space cannot be embedded
into any countably complete, countably upper continuous, and countably lower
continuous lattice. A similar result holds for the lattice of all order-convex
subsets of any infinite chain.Comment: To appear in Algebra Universali
The extreme vulnerability of interdependent spatially embedded networks
Recent studies show that in interdependent networks a very small failure in
one network may lead to catastrophic consequences. Above a critical fraction of
interdependent nodes, even a single node failure can invoke cascading failures
that may abruptly fragment the system, while below this "critical dependency"
(CD) a failure of few nodes leads only to small damage to the system. So far,
the research has been focused on interdependent random networks without space
limitations. However, many real systems, such as power grids and the Internet,
are not random but are spatially embedded. Here we analytically and numerically
analyze the stability of systems consisting of interdependent spatially
embedded networks modeled as lattice networks. Surprisingly, we find that in
lattice systems, in contrast to non-embedded systems, there is no CD and
\textit{any} small fraction of interdependent nodes leads to an abrupt
collapse. We show that this extreme vulnerability of very weakly coupled
lattices is a consequence of the critical exponent describing the percolation
transition of a single lattice. Our results are important for understanding the
vulnerabilities and for designing robust interdependent spatial embedded
networks.Comment: 13 pages, 5 figure
Sublattices of lattices of order-convex sets, I. The main representation theorem
For a partially ordered set P, we denote by Co(P) the lattice of order-convex
subsets of P. We find three new lattice identities, (S), (U), and (B), such
that the following result holds. Theorem. Let L be a lattice. Then L embeds
into some lattice of the form Co(P) iff L satisfies (S), (U), and (B).
Furthermore, if L has an embedding into some Co(P), then it has such an
embedding that preserves the existing bounds. If L is finite, then one can take
P finite, of cardinality at most , where n is the number of
join-irreducible elements of L. On the other hand, the partially ordered set P
can be chosen in such a way that there are no infinite bounded chains in P and
the undirected graph of the predecessor relation of P is a tree
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