No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional $\operatorname{iff}$ its theory has the independence property $\operatorname{iff}$ its theory has the multi-order (...) property. (shrink)

Every ℵ 0 -categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ 0 -categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ 0 -categorical partially ordered set not embedding one of them has a decidable theory.

Starting from a connected, partially ordered set of events, it is shown that results of the measurement of time are elements of a partially ordered and filtering field, as used in a previous paper. Moreover, some relations between physical formulas and properties of the field are proved. Finally, some open problems and suggestions are pointed out. For the convenience of the reader not acquainted with elementary algebraic methods, proofs are given in detail.

It is shown that under GCH every poset preserves its confinality in any cofinality preserving extension. On the other hand, starting with ω measurable cardinals, a model with a partial ordered set which can change its cofinality in a cofinality preserving extension is constructed.

A partially ordered set is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of . Assuming the set-theoretic assumption ⋄, it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not -homogeneous. This is impossible in the countable case for k ≥ 4.

In their paper from 1981, Milner and Sauer conjectured that for any poset $\langle P,\le\rangle$ , if $cf(P,\le)=\lambda>cf(\lambda)=\kappa$ , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known (...) axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown. (shrink)

This paper gives a new, proof-theoretic explanation of partial-order reasoning about time in a nonmonotonic theory of action. The explanation relies on the technique of lifting ground proof systems to compute results using variables and unification. The ground theory uses argumentation in modal logic for sound and complete reasoning about specifications whose semantics follows Gelfond and Lifschitz’s language. The proof theory of modal logic A represents inertia by rules that can be instantiated by sequences of time steps or events. Lifting (...) such rules introduces string variables and associates each proof with a set of string equations; these equations are equivalent to a set of partial-order tree-constraints that can be solved efficiently. The defeasible occlusion of inertia likewise imposes partial-order constraints in the lifted system. By deriving an auxiliary partial order representation of action from the underlying logic, not the input formulas or proofs found, this paper strengthens the connection between practical planners and formal theories of action. Moreover, the general correctness of the theory of action justifies partial-order representations not only for forward reasoning from a completely specified start state, but also for explanatory reasoning and for reasoning by cases. (shrink)

A partial ordering P is said to have the weak Freese-Nation property if there is a mapping tf : P → [P]0 such that, for any a, b ε P, if a b then there exists c ε tf∩tf such that a c b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, (...) every ccc complete Boolean algebra has this property while, under b 2, there exists no complete Boolean algebra with the WFN. (shrink)

This paper is primarily concerned with ℵ 0 -categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ 0 -categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ 0 -categorical U there is a linear order of A such that $(\mathfrak{U}, is ℵ 0 -categorical. Corollary 6.7. Every ℵ 0 -categorical theory of a partially ordered set of finite width (...) has a decidable theory. Theorem 7.7. Every ℵ 0 -categorical theory of reticles has a decidable theory. There is a section dealing just with decidability of partially ordered sets, the main result of this section being. Theorem 8.2. If $(P, is a finite partially ordered set and K P is the class of partially ordered sets which do not embed $(P, , then Th(K P ) is decidable iff K P contains only reticles. (shrink)

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is (...) the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper. (shrink)

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there (...) is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering ${\mathcal{A}}$ has the feature that for every ${\mathcal{B} \cong \mathcal{A}}$ , there is an infinite chain or antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , then we have uniform dichotomy: either for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite chain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , or for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$. (shrink)

J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability ? We show that having such an extension is a Σ 1 l -complete property and so there is no Borel answer to Łoś's question. Additionally, we show that there is a natural Π 1 l -norm on the partial orders which cannot be so extended and calculate some natural ranks (...) in that norm. (shrink)

As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on variable y, then (...) we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly logic. (shrink)

Various forms of mathematical induction are applicable to domains with some kinds of order. This naturally leads to the questions about the possibility of unification of different inductions and their generalization to wider classes of ordered domains. In the paper we propose a common framework for formulating induction proof principles in various structures and apply it to partially ordered sets. In this framework we propose a fixed induction principle which is indirectly applicable to the class of all (...) posets. In a certain sense, this result provides a solution to the problem of formulating a common generalization of a number of well-known induction principles for discrete and continuous ordered structures. (shrink)

This paper uses a partially ordered set of syntactic categories to accommodate optionality and licensing in natural language syntax. A complex but well-studied data set pertaining to the syntax of quantifier scope and negative polarity licensing in Hungarian is used to illustrate the proposal. The presentation is geared towards both linguists and logicians. The paper highlights that the main ideas can be implemented in different grammar formalisms, and discusses in detail an implementation where the partial ordering on categories (...) is given by the derivability relation of a calculus with residuated and Galois-connected unary operators. (shrink)

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$$\end{document} in question, d > 1, we fix a natural extension of Peano Arithmetic, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T \supseteq \sf{PA}}$$\end{document}, that proves the corresponding second-order sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{WPO}\left({\rm (...) S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$$\end{document}. Having this we consider the following parametrized first-order slow well-partial-ordering sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| 1}$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$$\end{document} is not provable in T.Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _{1}^{1}\sf{CA}}$$\end{document}, if d = 2 and d = 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$$\end{document} and prove analogous theorems. (In the sequel we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}_{+}}$$\end{document} the set of EFA-provably computable positive reals). In the basic case T = PA we strengthen the basic phase transition result by adding the following static threshold clause\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$$\end{document} is still provable in T = PA (actually in EFA). Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1α given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$$\end{document} being the familiar fast growing Hardy function and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$$\end{document} the corresponding slowly growing inversion. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha < \varepsilon _{0}}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$$\end{document} is provable in T = PA. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$$\end{document} is not provable in T = PA. We conjecture that this pattern is characteristic for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T\supseteq \sf{PA}}$$\end{document} under consideration and their proof-theoretical ordinals o (T ), instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon _{0}}$$\end{document}. (shrink)

Nondeterministic exponential time complexity bounds are established for recognizing true propositional formulas with partially ordered quantifiers on propositional variables.

Partial order optimality theory is a conservative generalization of classical optimality theory that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \ that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under some constraint set (...) C compatible with \?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C, for an arbitrary \, I provide set-theoretic means for constructing the actual set G compatible with \. Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \. As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT. (shrink)

The tree-property and its variants make sense also for directed sets and even for partially ordered sets. A combinatoria approach is developed here, with characterizations and criteria involving adequate families of special substructures of directed sets. These substructures form a natural hierarchy that is also investigated.

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)

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less (...) than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

We investigate first order sentences valid in completions of a given partial algebraic structure - a partial model. We give semantic and syntactic description of the set of all sentences valid in every completion of the given partial model - first order theory of this model.

In this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are (...) obtained for different kinds of Beck and Frobenius conditions. (shrink)

We articulate a relational understanding of modality, and show it at work in a survey of pre-order models, set-lifting, and modal languages.

In recent years, semantical partiality has emerged as an important explanatory concept in philosophical logic as well as in the study of natural language semantics. Despite the many applications, however, a number of mathematically intriguing questions associated with this concept have received only very limited attention. ;The present dissertation aims to present a systematic study of certain types of partiality in the area of basic model theory. Two types of issues are given special attention: Introducing partially defined models, there (...) are many ways to generalize the classical truth definition for sentences of a simple first order language relative to standard, complete models. Different interpretations of the formal framework motivate conflicting truth definitions between language and partial models: A partial model can be taken to represent a part of the world, or a partial information set. The truth of a sentence can be supported directly by a part of the world, but can also follow indirectly from an information set. These notions are related, and the relation motivates a comparison between various weaker and stronger alternative truth definitions. Results are obtained about the extent to which these truth definitions differ, and a number of characterization results are deduced. ;Among other conditions that are not expressible in the framework of standard, complete model theory, a condition of monotonicity or persistence of truth relative to partial models is argued to follow under both the given interpretations of the formal framework. The final chapter investigates the relation between such conditions and expressibility properties in general. These discussions culminate with a combined Lindstrom and persistence characterization theorem. (shrink)

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, (...) proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems. (shrink)

For f: On → On let supp: = ξ: 0, and let S := {f : On → On : supp finite}. For f,g ϵ S definef ≤ g : ↔ [h one-to-one ⁁ f ≤ g)].A function ψ : S → On is called monotonic increasing, if f≤ψ and if f ≤ g implies ψ ≤ ψ. For a mapping ψ : S → On let Clψ be the least set T of ordinals which contains 0 as an element (...) and which is closed under the following rule: If f ϵ S, range ⊆ T and supp ⊆ T, then ψ ϵ T. Let φ be the enumeration function of the class {ξ: [ξ = ωn]}. Let φ be the induced Schütte-Klammerterm-function which generates the Schütte-Veblen-hierarchy of ordinals. Then φ is monotonic increasing. We show: If ψ: S → On is monotonic increasing, then otyp) ≤ min {ξ : ξ = Φ, where 1ξ = 1 if α = ξ and 1ξ = 0 if α ↔ = 0 if α ≠ ξ. For τ an ordinal let Sτ := {f ϵ S : supp ⊆ τ}. A function ψ : Sτ → On is called τ-monotonic increasing if fξ and if f ≤ g implies ψ ≤ . For a function ψ Sτ → On let Clψ ↔ τ be the least set T of ordinals, which contains 0 as an element, such that if f ϵ Sτ and range ϵ T, then ψ ϵ T. We show: if ≥2 and if ψ Sτ → On is τ-monotonic increasing, then otyp ↔ τ) ≤. We also prove a generalization of this theorem in terms of well-partial orderings. MSC: 03F15, 03E10, 06A06. (shrink)

We employ the notions of "sequential function" and "interrogation" (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is that every realizability topos is a geometric quotient of a realizability topos on a total combinatory algebra.

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly (...) class='Hi'>ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

Weakly semirecursive sets have been introduced by Jockusch and Owings . In the present paper their investigation is pushed forward by utilizing r.e. partial orderings, which turn out to be instrumental for the study of degrees of subclasses of weakly semirecursive sets.

We propose a model of unawareness that remains close to the paradigm of Aumann’s model for knowledge [R. J. Aumann, International Journal of Game Theory 28 (1999) 263-300]: just as Aumann uses a correspondence on a state space to define an agent’s knowledge operator on events, we use a correspondence on a state space to define an agent’s awareness operator on events. This is made possible by three ideas. First, like the model of [A. Heifetz, M. Meier, and B. Schipper, (...) Journal of Economic Theory 130 (2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107- 124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well- known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences. (shrink)

In the paper it is shown that every physically sound Birkhoff – von Neumann quantum logic, i.e., an orthomodular partially ordered set with an ordering set of probability measures can be treated as partial infinite-valued Łukasiewicz logic, which unifies two competing approaches: the many-valued, and the two-valued but non-distributive, which have co-existed in the quantum logic theory since its very beginning.

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all (...) of these models, the algebraic closure of a set is nontrivial., is given in §1, however in vector spaces, cl is just the subspace generated byS, in Boolean algebras, cl is just the subalgebra generated byS, and in algebraically closed fields, cl is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings. (shrink)

This paper is about partially ordered multisets (pomsets for short). We investigate a particular class of pomsets that we call order-deterministic, properly including all partially ordered sets, which satisfies a number of interesting properties: among other things, it forms a distributive lattice under pomset prefix (hence prefix closed sets of order-deterministic pomsets are prime algebraic), and it constitutes a reflective subcategory of the category of all pomsets. For the order-deterministic pomsets we develop an algebra with a (...) sound and (-) complete equational theory. The operators in the algebra are concatenation and join, the latter being a variation on the more usual disjoint union of pomsets. This theory is then extended in order to capture refinement of pomsets by incorporating homomorphisms between models as objects in the algebra and homomorphism application as a new operator. (shrink)

One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame (...) of the logic and therefore characterises orthomodular quantum logic. To prove soundness, a further restriction is placed on admissible valuations which requires that they be point-generated in addition to the requirement that they return stable sets. (shrink)

We propose a notion of -minimality for partially ordered structures. Then we study -minimal partially ordered structures such that is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize -categoricity in their setting. Finally, we classify -minimal Boolean algebras as well as -minimal measure spaces.

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...) In this paper, we provide partial answers to the above open problems and among other results, we establish the following:1.For every integer \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \. (shrink)