1,312 research outputs found
A scattering of orders
A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in . More generally, we say that a partial ordering is -scattered if it does not contain a copy of any -dense linear ordering. We prove analogues of Hausdorff's result for -scattered linear orderings, and for -scattered partial orderings satisfying the finite antichain condition. We also study the -scattered partial orderings, where is the saturated linear ordering of cardinality , and a partial ordering is -scattered when it embeds no copy of . We classify the -scattered partial orderings with the finite antichain condition relative to the -scattered linear orderings. We show that in general the property of being a -scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders
We investigate the ordinal invariants height, length, and width of well quasi
orders (WQO), with particular emphasis on width, an invariant of interest for
the larger class of orders with finite antichain condition (FAC). We show that
the width in the class of FAC orders is completely determined by the width in
the class of WQOs, in the sense that if we know how to calculate the width of
any WQO then we have a procedure to calculate the width of any given FAC order.
We show how the width of WQO orders obtained via some classical constructions
can sometimes be computed in a compositional way. In particular, this allows
proving that every ordinal can be obtained as the width of some WQO poset. One
of the difficult questions is to give a complete formula for the width of
Cartesian products of WQOs. Even the width of the product of two ordinals is
only known through a complex recursive formula. Although we have not given a
complete answer to this question we have advanced the state of knowledge by
considering some more complex special cases and in particular by calculating
the width of certain products containing three factors. In the course of
writing the paper we have discovered that some of the relevant literature was
written on cross-purposes and some of the notions re-discovered several times.
Therefore we also use the occasion to give a unified presentation of the known
results
Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings
We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations ρ1,...,ρk, is the downward closed set Av(ρ1,...,ρk) consisting of all equivalence relations which do not contain any of ρ1,...,ρk (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?Peer reviewe
Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings
We consider the posets of equivalence relations on finite sets under the
standard embedding ordering and under the consecutive embedding ordering. In
the latter case, the relations are also assumed to have an underlying linear
order, which governs consecutive embeddings. For each poset we ask the well
quasi-order and atomicity decidability questions: Given finitely many
equivalence relations , is the downward closed set
Av consisting of all equivalence relations which do not
contain any of : (a) well-quasi-ordered, meaning that it
contains no infinite antichains? and (b) atomic, meaning that it is not a union
of two proper downward closed subsets, or, equivalently, that it satisfies the
joint embedding property
A Few Notes on Formal Balls
Using the notion of formal ball, we present a few new results in the theory
of quasi-metric spaces. With no specific order: every continuous
Yoneda-complete quasi-metric space is sober and convergence Choquet-complete
hence Baire in its -Scott topology; for standard quasi-metric spaces,
algebraicity is equivalent to having enough center points; on a standard
quasi-metric space, every lower semicontinuous -valued
function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the
continuous Yoneda-complete quasi-metric spaces are exactly the retracts of
algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete
quasi-metric space has a so-called quasi-ideal model, generalizing a
construction due to K. Martin. The point is that all those results reduce to
domain-theoretic constructions on posets of formal balls
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