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Let OML denote the class of orthomodular lattices (OMLs, quantum logics). Let L be an OML and let B be a maximal Boolean subalgebra of L. Then B is called a block of L. In the algebraic investigation of OMLs a natural question is whether the blocks of a product (resp. ultraproduct) of OMLs are products (resp. ultraproducts) of the blocks of the respective “coordinate” OMLs. We first add to the study of this question as regards the products and the centres of the products (a special mention deserves the result that the centre of the ultraproduct is the ultraproduct of the centres of the respective OMLs). Then we pass to the analogous questions for ultraproducts where we present main results of this note. Though this question on the “regular” behaviour of blocks in ultraproducts remains open in general, we provide a positive partial solution. This contributes to the understanding of varieties important to quantum theories – to the varieties that contain both set-representable OMLs and projection OMLs. We consider an axiomatizable class of the OMLs, OML n , whose blocks uniformly intersect in finite sets of the maximal cardinality of 2 n . It is worth realizing within the connection to quantum logic theory that, for instance, the OMLs given by Greechie diagrams belong to OML 2 . The importance of the results is commented on in relation to the state space properties of OMLs. [ABSTRACT FROM AUTHOR]

We continue the investigation started in Golshani (2021) about the relation between the Keilser–Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence m = ⟨ (M n 1 , M n 2) : n < ω ⟩ of models of size at most ℵ 1 in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter D on ω , if the ultraproducts ∏ D M n 1 and ∏ D M n 2 are elementarily equivalent, then they are isomorphic. [ABSTRACT FROM AUTHOR]

In this work, we survey some research directions in which the ultraproduct construction and methods based on ultrafilters play significant roles. Rather different areas of mathematics have been considered: topics we are reviewing here include some aspects of the model theory of first-order and second-order existential logics, finite Ramsey theory and general topology. Special emphasis has been made for producing a uniform treatment and for highlighting interconnections between these different subjects. [ABSTRACT FROM AUTHOR]

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if μ is definable and both μ and ν are pseudo-finite, then the Morley product of μ and ν agrees with the pseudo-finite product of μ and ν. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups. [ABSTRACT FROM AUTHOR]

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct L ⁎ has, as a spine, η 1 , an uncountable analogue of the order type of rationals η. Finite big Ramsey degrees for η were exactly calculated by Devlin in [5]. It is immediate from [39] that η 1 fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to η 1 to show that it witnesses big Ramsey degrees of finite tuples in η on every copy of η in η 1 , and consequently in L ⁎. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures. [ABSTRACT FROM AUTHOR]

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring F q [ t ] over a finite field F q that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p > 0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and p > 0 for some prime p. [ABSTRACT FROM AUTHOR]

In this work, we survey some research directions in which the ultraproduct construction and methods based on ultrafilters play significant roles. Rather different areas of mathematics have been considered: topics we are reviewing here include some aspects of the model theory of first-order and second-order existential logics, finite Ramsey theory and general topology. Special emphasis has been made for producing a uniform treatment and for highlighting interconnections between these different subjects.

We study properties of a group, abelian group, ring, or monoid B which (a) guarantee that every homomorphism from an infinite direct product ΠI Ai of objects of the same sort onto B factors through the direct product of finitely many ultraproducts of the Ai (possibly after composition with the natural map B → B/Z(B) or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions and topics for further investigation are noted.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields. [ABSTRACT FROM AUTHOR]

A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions. [ABSTRACT FROM AUTHOR]

We study geometric properties of GL-spaces. We demonstrate that every finite-dimensional GL-space is polyhedral; that in dimension 2 there are only two, up to isometry, GL-spaces, namely the space whose unit sphere is a square (like ℓ ∞ 2 or ℓ 1 2 ) and the space whose unit sphere is an equilateral hexagon. Finally, we characterise the spaces E = (R n , ‖ · ‖ E) with absolute norm such that for every collection X 1 , ... , X n of GL-spaces their E-sum is a GL-space. [ABSTRACT FROM AUTHOR]

This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a algebra is a very interesting problem in itself. Considering the ultraproducts of -algebras and the states on these algebras, we get quite nontrivial results. [ABSTRACT FROM AUTHOR]

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields. [ABSTRACT FROM AUTHOR]

Let S be a Noetherian scheme and ƒ : X → S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules Ri ƒ*F obtained by direct image (for the étale topology) are themselves constructible, that is, there is a stratification S of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of S on F. We show that a natural ‘uniform’ tameness and constructibility condition satisfied by constant sheaves, which was introduced by O. Gabber, is stable under the functors Ri ƒ*. If f is not proper, this result still holds assuming tameness at infinity, relative to S. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the sheaves Ri ƒ*Fl, where L ranges through all prime numbers invertible on S. [ABSTRACT FROM AUTHOR]

Quantum Kripke frames are certain quantum structures recently introduced by Zhong. He has defined certain properties such as Existence of Approximation and Superposition for these structures. In this paper, we define the ultraproduct for the family of quantum Kripke frames and show that the aforementioned properties are invariant under ultraproduct. In this way we prove that the ultraproduct of each family of quantum Kripke frames is also a quantum Kripke frame. We also show the same results for other related quantum structures. [ABSTRACT FROM AUTHOR]

Abstract Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results. [ABSTRACT FROM AUTHOR]

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. [ABSTRACT FROM AUTHOR]

Cerreia-Vioglio et al. (Econ Theory 48(2-3):341-375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed. [ABSTRACT FROM AUTHOR]

We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of ultraproducts of (finite) cardinals modulo D which are infinite. [ABSTRACT FROM AUTHOR]

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction. [ABSTRACT FROM AUTHOR]

Let m and n be cardinals with 3 ≤ m , n ≤ ω . We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than m and n respectively cannot be finitely axiomatized. [ABSTRACT FROM AUTHOR]

We continue our study of the class ${\cal C}\left( D \right)$ , where D is a uniform ultrafilter on a cardinal κ and ${\cal C}\left( D \right)$ is the class of all pairs $\left( {{\theta _1},{\theta _2}} \right)$ , where $\left( {{\theta _1},{\theta _2}} \right)$ is the cofinality of a cut in ${J^\kappa }/D$ and J is some ${\left( {{\theta _1} + {\theta _2}} \right)^ + }$ -saturated dense linear order. We give a combinatorial characterization of the class ${\cal C}\left( D \right)$. We also show that if $\left( {{\theta _1},{\theta _2}} \right) \in {\cal C}\left( D \right)$ and D is ${\aleph _1}$ -complete or ${\theta _1} + {\theta _2} > {2^\kappa }$ , then ${\theta _1} = {\theta _2}$. [ABSTRACT FROM AUTHOR]

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ 1 ( N 0 ) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ 1 ( Z ) nor in ℓ 1 ( Q ) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed. [ABSTRACT FROM AUTHOR]

By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on q -Gaussian algebras transfer to q -Araki–Woods algebras. As a consequence, we establish the w ⁎ -complete metric approximation property for all q -Araki–Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C ⁎ -subalgebras of q -Araki–Woods algebras are always QWEP. [ABSTRACT FROM AUTHOR]

We give a complete characterization of torsion-complete and algebraically compact abelian groups whose endomorphism rings admit a compact ring topology. [ABSTRACT FROM PUBLISHER]

An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties. [ABSTRACT FROM AUTHOR]

In this paper, we first study the fixed point property for nonexpansive mappings of a Banach space and some existing result in [7] is extended. We secondly study the relationship between uniform normal structure and slices and some results in [11] are improved too. [ABSTRACT FROM AUTHOR]

We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and σ-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ-algebra defined by a collection of subsets of [ABSTRACT FROM AUTHOR]

For an ultrafilter on a cardinal we wonder for which pair of regular cardinals, we have: for any -saturated dense linear order has a cut of cofinality We deal mainly with the case [ABSTRACT FROM AUTHOR]

We find some sufficient conditions for a radical class of an idempotent radical in the category of modules over a Dedekind left bounded duo-domain to be axiomatizable. In the case of the integer numbers ring this result implies the Gorbachuk-Komarnitskii Theorem on axiomatizable radical classes of Abelian groups.