Two-dimensional partial orderings: Recursive model theory

The uniform content of partial and linear orders

§1. Introduction. Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets) always seem to be either actually computable (recursive) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?Let be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl)with least element 0, the degree (equivalence class) of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of .The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:Theorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into .Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b.

Chapter 10 Complexity theoretic model theory and algebra

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

The solution of equations and systems of equations over real, complex, rational and integer numbers is a classic topic of research in various areas of mathematics for several thousand years. In the last 20 years, the so-called universal algebraic geometry has been actively developed, in which systems of equations over arbitrary algebraic systems are researched. Many practically important problems on finite graphs, finite fields, and finite orders can be formulated as problems related to solving systems of equations over these systems, which leads to the need to develop algebraic geometry. Many modern models of informational defence represents by graphs and partial orders (posets). This article presents polynomial algorithms for constructing radical and coordinate partial order of systems of equations over finite partial orders in a language without constants.

An application of the theory of categorical many-valued partial orders to sets with fuzzy equalities

In Chapter 3, we have seen how averaged heights of objects hav(x) can be calculated. Thus partial order provides a method to obtain a linear order without the need of making additional assumptions like weights for indicators. The main computational problem, however, is the huge number of linear extensions, which sometimes makes the calculation of averaged heights and from them the linear order difficult. This chapter discusses different procedures to rank objects.

E. Szpilrajn [2] showed that any partial order r on a set A can be extended to a linear order on A. He also proved that any partial order is the intersection of its linear extensions. The following generalization of Szpilrajn's classical result can be found in [1]. Given a poset (A, r) and an acyclic r-monotone function f : A-~A, then r can be extended to a linear order R with xRy=~f(x)RJ(y) for all x, yCA. Such linear extensions are called monotonicity preserving or simply f-linear extensions. The triple (A,f , R) can be considered as a linearly ordered extension of the acyclic partially ordered 1-unary algebra (A,f, r). The main aim of this note is to describe the intersection of all f-linear extensions of a given acyclic partially ordered algebra (A,f, r). In what follows we present an improved version of a theorem first a n n o u n c e d in [1].

Forbidden induced partial orders

In many of the possible applications as well as the theoretical models of computational social choice, the agents' preferences are represented as partial orders. In this paper, we extend the maximum likelihood approach for defining optimal voting rules to this setting. We consider distributions in which the pairwise comparisons/incomparabilities between alternatives are drawn i.i.d. We call such models pairwise-independent models and show that they correspond to a class of voting rules that we call pairwise scoring rules. This generalizes rules such as Kemeny and Borda. Moreover, we show that Borda is the only pairwise scoring rule that satisfies neutrality, when the outcome space is the set of all alternatives. We then study which voting rules defined for linear orders can be extended to partial orders via our MLE model. We show that any weakly neutral outcome scoring rule (including any ranking/candidate scoring rule) based on the weighted majority graph can be represented as the MLE of a weakly neutral pairwise-independent model. Therefore, all such rules admit natural extensions to profiles of partial orders. Finally, we propose a specific MLE model πk for generating a set of k winning alternatives, and study the computational complexity of winner determination for the MLE of πk.

Balanced pairs in partial orders

Primary facets of order polytopes

We find out suitable conditions on a To-principal topology, under which the associated partial order is a partial order with nontransitive incomparability, that is an interval order, a partial semiorder or a semiorder. In order to perform these characterizations, only a T 1 separation axiom is needed. The settheoretical approach allows us to give a simple proof of a fundamental theorem due to Fish burn, concerning the numerical representation of interval orders. We also introduce a class of planar interval orders, called strong interval orders. Although planar posets, as well as interval orders, have arbitrary finite dimension, we prove that a strong interval order is the intersection of at most two linear orders.

This paper calculates, in a precise way, the complexity of the index sets for computable structures that are autostable relative to strong constructivizations and belong to one of the following classes: linear orderings, Boolean algebras, distributive lattices, partial orderings, rings, and commutative semigroups. We also calculate the complexity of the index set for computable structures that have computable dimension n, where n is a fixed natural number greater than 1.

Discussion of the relation between exchange-value and use-value (as defined inCapital I) is clarified by the construction of set-theoretical models of these concepts. Marx argues fallaciously for the independence of exchange-value and use-value. His fallacy is diagnosed as depending upon a mistaken assumption about the impossibility of inferring a certain linear order on a set from a certain (different) partial order on that set.