In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics.

In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the -completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann , Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384–389]. A key result is that all completions of a certain type are (...) universal, hence unique . We give a direct definition of the -completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the -completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the -completion agrees with the sobrification respectively the closed ideal completion. (shrink)

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames. It was established in [ Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in the two-sorted classical predicate (...) calculusCPC 2. Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB n P m + characterized by the class of all posets of the finite height m and the finite branching n. A finite axiomatization of the predicate logicLP m + characterized by the class of all posets of the height m is known from [Yokota 1989]. We prove thatLB n P m + =,B n being the propositional axiom of the branching n. (shrink)

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in §2 below.

A D-algebra is a generalization of a D-poset in which a partial order is not assumed. However, if a D-algebra is equipped with a natural partial order, then it becomes a D-poset. It is shown that D-algebras and effect algebras are equivalent algebraic structures. This places the partial operation ⊝ for a D-algebra on an equal footing with the partial operation ⊕ for an effect algebra. An axiomatic structure called an effect stale-space is introduced. Such spaces provide an (...) operational interpretation for the partial operations ⊕ and ⊝. Finally, a relationship between effect-state spaces and torsion free interval effect algebras is demonstrated. (shrink)

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called equation image without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in (...) this paper a Kripke-style semantics for a subsystem equation image of equation image for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system equation image with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic equation image. We introduce a reational semantics which has the advantage to allow an interpretation of the propositionnal logic equation image using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic equation image. (shrink)

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called equation image without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in (...) this paper a Kripke-style semantics for a subsystem equation image of equation image for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system equation image with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic equation image. We introduce a reational semantics which has the advantage to allow an interpretation of the propositionnal logic equation image using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic equation image. (shrink)

We investigate the partial orderings of the form 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}(\mathbb{X}), \subset \rangle}$$\end{document}, where X=〈X,ρ〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X} =\langle X, \rho \rangle }$$\end{document} is a countable binary relational structure and P(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} (\mathbb{X})}$$\end{document} the set of the domains of its isomorphic substructures and show that if the components of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} (...) are maximally embeddable and satisfy an additional condition related to connectivity, then the poset 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P} (\mathbb{X}), \subset \rangle }$$\end{document} is forcing equivalent to a finite power of (P(ω)/ Fin)+, or to the poset (P(ω × ω)/(fin × Fin))+, or to the product (P(Δ)/EDfin)+×((P(ω)/Fin)+)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(P(\Delta )/\fancyscript{e}\fancyscript{d}_{\rm fin})^+ \times ((P(\omega )/{\rm Fin})^+)^n}$$\end{document}, for some n∈ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings. (shrink)

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a (...) non-reflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [math]-Prikry. We showed that given a [math]-Prikry poset [math] and a [math]-name for a (...) non-reflecting stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In this paper, we develop a general scheme for iterating [math]-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. (shrink)

A. M. Pitts in [Pi] proved that HA op fp is a bi-Heyting category satisfying the Lawrence condition. We show that the embedding $\Phi: HA^\mathrm{op}_\mathrm{fp} \longrightarrow Sh(\mathbf{P_0,J_0})$ into the topos of sheaves, (P 0 is the category of finite rooted posets and open maps, J 0 the canonical topology on P 0 ) given by $H \longmapsto HA(H,\mathscr{D}(-)): \mathbf{P_0} \longrightarrow \text{Set}$ preserves the structure mentioned above, finite coproducts, and subobject classifier, it is also conservative. This whole structure on HA op (...) fp can be derived from that of Sh(P 0 ,J 0 ) via the embedding Φ. We also show that the equivalence relations in HA op fp are not effective in general. On the way to these results we establish a new kind of duality between HA op fp and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi]. (shrink)

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$, the $(n+2)$ -Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$ -paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

Does every partial order of singular cofinality λ have an antichain of size ? This is the Singular Cofinality Conjecture. M. Pouzet proved [M. Pouzet, Parties cofinales des ordres partiels ne contenant pas d’antichaines infinies, 1980, preprint] that there must be an infinite antichain. When is uncountable, the positive answer is only consistently true, but unknown in ZFC. In this note we investigate this question from the purely set-theoretic point of view. On the way, we answer a question of Milner (...) and Pouzet from [E.C. Milner, M. Pouzet, Posets with singular cofinality, 1997, preprint]. (shrink)

The age of a relational structure R is the set A of finite restrictions of R considered up to isomorphism. R. Fraı̈ssé, who introduced this notion, showed that ages coincide with nonempty ideals of the poset consisting of finite relational structures, considered up to isomorphism and ordered by embeddability. Here, given two ages A ⊆ B , we study the poset D consisting of ages C in sandwich between A and B . Among other things we show that (...) if D is infinite then it contains an infinite chain. (shrink)

In this important study D. M. Armstrong offers a comprehensive system of analytical metaphysics that synthesises but also develops his thinking over the last twenty years. Armstrong's analysis, which acknowledges the 'logical atomism' of Russell and Wittgenstein, makes facts the fundamental constituents of the world, examining properties, relations, numbers, classes, possibility and necessity, dispositions, causes and laws. All these, it is argued, find their place and can be understood inside a scheme of states of affairs. This is a comprehensive and (...) rigorously this-worldly account of the most general features of reality, argued from a distinctive philosophical perspective, and it will appeal to a wide readership in analytical philosophy. (shrink)

The EU Kids Online project aims to enhance knowledge of the experiences and practices of European children and their parents regarding online risks and safety. A crucial research effort by the EU Kids Online network has been a survey in 25 European countries which targeted approximately 1,000 children per country. This article applies a cross-cultural values filter to the data that were gathered on parental mediation and the Internet in this survey. Our intention is to test whether Geert Hofstede’s cross-national (...) research results about national cultural values also apply to the EU Kids Online data on parental mediation. This implies studying collectivism versus individualism, low versus high power distance, masculinity versus femininity and low versus high uncertainty avoidance. We test whether differences between nations on these four dimensions correlate with differences between countries in parental mediation of the Internet and we test which European countries form clusters. (shrink)

This article reports on a survey of youth in Brussels and their ownership and use of digital technologies, focusing specifically on the social and cultural diversity within this group. Socio-cultural diversity includes differences regarding ethnicity and gender, language and educational attainment, as well as social and economic status. The relationship of these socio-cultural differences with the digital divide in terms of ownership and use is investigated. The data show a persistent ownership divide between socially weaker versus stronger groups accompanied by (...) an emerging socially compensatory function of instrumentally oriented user profiles among the socially weaker groups. (shrink)

The concept of human dignity is widely used in contemporary ethics and law as a foundational criterion for moral reasoning. Nonetheless, the concept has recently received criticism from various quarters. Some of this criticism has come from representatives of the animal liberation movement. The concept of human dignity is accused of underpinning an ethics that is anthropocentric and speciesist. That is, human dignity is said to be used as the basis of an ultimately unjustifiable attribution of intrinsic moral worth only (...) to human beings and to lead, consequently, to a detrimental prejudice against other species. (shrink)

It is a truth of military history that major battles are not clearly understood by the rank-and-file who are embroiled in them. There is a flurry of activity, a “blooming, buzzing confusion,” and anything that moves in the surrounding terrain is likely to be identified as the enemy. Usually it is only after the “tumult and the shouting dies” that a clear picture emerges, and a tally can be obtained of how many of one’s friends were felled by mistake.