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In this article we show that biintuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication
$\phi \rightarrow \psi $
with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for biintuitionistic logic with global consequence, the two formulations of the property are not equivalent.
We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub
$\mathtt {ML}^{\infty }$
and
$\mathtt {GML}^{\infty }$
, respectively. We then characterise these logics as the bisimulationinvariant fragments of the naturally corresponding predicate logic, viz., the extension of firstorder logic with the infinity quantifier. Furthermore, for both
$\mathtt {ML}^{\infty }$
and
$\mathtt {GML}^{\infty }$
we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.
We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the pprimary part of A is either finite or it coincides with the Prüfer pgroup. We also provide a modeltheoretic description of the model companions we obtain.
We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and MoosaPillay and show that they do not hold in the counterexample. In order to do so, a study of imaginaries in additive covers is developed. As a byproduct of the presentation, we observe that a pure bindinggrouptheoretic account of the CBP is unlikely.
Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finitedimensional ${\boldsymbol {k}}$linear subspace of that couple. It follows that the differentialvalued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a twosorted structure with $\mathbb {R}$ as the underlying set for the second sort.
The complete characterisation of order types of nonstandard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their nonstandard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.
Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by
$\pi $
. Let
$\Lambda $
be an Rorder such that
$Q\Lambda $
is a separable Qalgebra. Maranda showed that there exists
$k\in \mathbb {N}$
such that for all
$\Lambda $
lattices L and M, if
$L/L\pi ^k\simeq M/M\pi ^k$
, then
$L\simeq M$
. Moreover, if R is complete and L is an indecomposable
$\Lambda $
lattice, then
$L/L\pi ^k$
is also indecomposable. We extend Maranda’s theorem to the class of Rreduced Rtorsionfree pureinjective
$\Lambda $
modules.
As an application of this extension, we show that if
$\Lambda $
is an order over a Dedekind domain R with field of fractions Q such that
$Q\Lambda $
is separable, then the lattice of open subsets of the Rtorsionfree part of the right Ziegler spectrum of
$\Lambda $
is isomorphic to the lattice of open subsets of the Rtorsionfree part of the left Ziegler spectrum of
$\Lambda $
.
Furthermore, with k as in Maranda’s theorem, we show that if M is Rtorsionfree and
$H(M)$
is the pureinjective hull of M, then
$H(M)/H(M)\pi ^k$
is the pureinjective hull of
$M/M\pi ^k$
. We use this result to give a characterization of Rtorsionfree pureinjective
$\Lambda $
modules and describe the pureinjective hulls of certain Rtorsionfree
$\Lambda $
modules.
Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows us to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.
We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All ominimal structures on
$(\mathbb {R},<)$
have the property, as do all expansions of
$(\mathbb {R},+,\cdot ,\mathbb {N})$
. Our main analyticgeometric result is that any such expansion of
$(\mathbb {R},<,+)$
by Boolean combinations of open sets (of any arities) either is ominimal or defines an isomorph of
$(\mathbb N,+,\cdot )$
. We also show that any given expansion of
$(\mathbb {R}, <, +,\mathbb {N})$
by subsets of
$\mathbb {N}^n$
(n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.
We provide a model theoretical and tree propertylike characterization of
$\lambda $

$\Pi ^1_1$
subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.
We introduce and study modeltheoretic connected components of rings as an analogue of modeltheoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use modeltheoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group
${\mathrm {UT}}_3({\mathbb {Z}})$
, the continuous Heisenberg group
${\mathrm {UT}}_3({\mathbb {R}})$
, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.
In this paper, using a propositional modal language extended with the window modality, we capture the firstorder properties of various mereological theories. In this setting,
$\Box \varphi $
reads all the parts (of the current object) are
$\varphi $
, interpreted on the models with a wholepart binary relation under various constraints. We show that all the usual mereological theories can be captured by modal formulas in our language via frame correspondence. We also correct a mistake in the existing completeness proof for a basic system of mereology by providing a new construction of the canonical model.
The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra
$\mathcal {B}$
to each formula. We show some basic results regarding the effect of the properties of
$\mathcal {B}$
on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.
We show that if a countable structure M in a finite relational language is not cellular, then there is an agepreserving
$N \supseteq M$
such that
$2^{\aleph _0}$
many structures are biembeddable with N. The proof proceeds by a case division based on mutual algebraicity.
Necessary and sufficient conditions are presented for the (firstorder) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes latticeordered abelian groups and MValgebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of latticeordered abelian groups and MValgebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including latticeordered abelian groups and MValgebras, can be extended with a binary operation to obtain theories that do have a model completion.
Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and logheight of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$. We prove effective bounds on the geometry of the intersection ${\mathcal B}\cap V$. In particular, when $\operatorname {codim} V=\dim {\mathcal L}$ we prove that $\#({\mathcal B}\cap V)$ is bounded by a polynomial in $\delta _{V}$ and $\log \operatorname {dist}^{1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${\mathcal B}\cap V$ by an algebraic map $\Phi $. For instance, under suitable conditions we show that $\Phi ({\mathcal B}\cap V)$ contains at most $\operatorname {poly}(g,h)$ algebraic points of logheight h and degree g.
We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galoisorbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.
We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of ominimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of tstratifications in Hensel minimal structures and Taylor approximation results that are key to nonArchimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equicharacteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.
We introduce and prove the consistency of a new set theoretic axiom we call the Invariant Ideal Axiom. The axiom enables us to provide (consistently) a full topological classification of countable sequential groups, as well as fully characterize the behavior of their finite products.
We also construct examples that demonstrate the optimality of the conditions in IIA and list a number of open questions.
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on HrushovskiLoeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in ominimal expansions of the tropical semigroup
$\Gamma _{\infty }$
, where
$\Gamma $
denotes the value group of K. For quasiprojective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of
$\Gamma _{\infty }$
. In both contexts, we show that the corresponding cohomology theory satisfies the EilenbergSteenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.
We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by
$H(F)$
the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in
$H(F)$
, using existential formulas with an arbitrary noncommuting pair of elements as parameters. We show that F is interpreted in
$H(F)$
using computable
$\Sigma _1$
formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of HarrisonTrainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in
$H(F)$
of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in
$H(F)$
. Looking at what was used to arrive at this parameterfree interpretation of F in
$H(F)$
, we give general conditions sufficient to eliminate parameters from interpretations.