194 research outputs found
Chains, Antichains, and Complements in Infinite Partition Lattices
We consider the partition lattice on any set of transfinite
cardinality and properties of whose analogues do not hold
for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the
cardinality of any maximal well-ordered chain is always exactly ; (II)
there are maximal chains in of cardinality ; (III) if,
for every cardinal , we have , there
exists a maximal chain of cardinality (but ) in
; (IV) every non-trivial maximal antichain in has
cardinality between and , and these bounds are realized.
Moreover we can construct maximal antichains of cardinality for any ; (V) all cardinals of the form
with occur as the number of
complements to some partition , and only these
cardinalities appear. Moreover, we give a direct formula for the number of
complements to a given partition; (VI) Under the Generalized Continuum
Hypothesis, the cardinalities of maximal chains, maximal antichains, and
numbers of complements are fully determined, and we provide a complete
characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201
Out of Nowhere: Spacetime from causality: causal set theory
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of
Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and
Christian W\"uthrich and under contract with Oxford University Press. (More
information at www.beyondspacetime.net.) This chapter introduces causal set
theory and identifies and articulates a 'problem of space' in this theory.Comment: 29 pages, 5 figure
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