Epimorphisms between linear orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the paragraphs introducing the results of subsection 3.3 (statements and proofs are unchanged

Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American Mathematical Societ

Lebesgue density and exceptional points

Work in the measure algebra of the Lebesgue measure on N2 : for comeagre many [A] the set of points x such that the density of x in A is not defined is \u3a30 3-complete; for some compact K the set of points x such that the density of x in K exists and it is different from 0 or 1 is \u3a00 3-complete; the set of all [K] with K compact is \u3a00 3-complete. There is a set (which can be taken to be open or closed) in \u211d such that the density of any point is either 0 or 1, or else undefined. Conversely, if a subset of \u211dn is such that the density exists at every point, then the value 1/2 is always attained on comeagre many points of the measurable frontier. On the route to these results we show that the Cantor space can be embedded in a measured Polish space in a measure-preserving fashio

Erratum to: Epimorphisms Between Linear Orders

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Modal operators and toric ideals

open3In the present paper, we consider modal propositional logic and look for the constraints that are imposed to the propositions of the special type Box by the structure of the relevant finite Kripke frame. We translate the usual language of modal propositional logic in terms of notions of commutative algebra, namely polynomial rings, ideals and bases of ideals. We use extensively the perspective obtained in previous works in algebraic statistics. We prove that the constraints on Box can be derived through a binomial ideal containing a toric ideal and we give sufficient conditions under which the toric ideal, together with the fact that the truth values are in 0,1, fully describes the constraints.openR. Camerlo, G. Pistone, F. RapalloCamerlo, R.; Giovanni, Pistone; Rapallo, F

Linear orders: When embeddability and epimorphism agree

When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is a D\u2c72(\u3a011)-complete set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders

Fences, their endpoints, and projective Fra\"iss\'e theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families $\mathcal F, \mathcal F_0$ of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from $\mathcal F_0$. We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that $\mathcal F, \mathcal F_0$ are projective Fra\"iss\'e families with a common projective Fra\"iss\'e limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fra\"iss\'e fence. We show that the Fra\"iss\'e fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fra\"iss\'e theory.Comment: Version accepted for publication in the Transaction of the American Mathematical Societ

On isometry and isometric embeddability between ultrametric Polish spaces

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length $\omega_1$ which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least $\omega+3$. We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.Comment: Minor imprecisions corrected. Following the suggestion of the anonymous referee, the former Section 7 concerning arbitrary Polish spaces with fixed set of distances has been left out and will be posted as a separate paper soon. The title is changed as well to reflect this modificatio