Additional Axioms Research Articles

Completeness Theorems for Kleene algebra with tests and top

We prove two completeness results for Kleene algebra with tests and a top element, with respect to guarded string languages and binary relations. While the equational theories of those two classes of models coincide over the signature of Kleene algebra, this is no longer the case when we consider an additional constant ``top'' for the full element. Indeed, the full relation satisfies more laws than the full language, and we show that those additional laws can all be derived from a single additional axiom. We recover that the two equational theories coincide if we slightly generalise the notion of relational model, allowing sub-algebras of relations where top is a greatest element but not necessarily the full relation. We use models of closed languages and reductions in order to prove our completeness results, which are relative to any axiomatisation of the algebra of regular events. For one of our constructions, we extend the concept of finite monoid recognisability to guarded-string languages; this device makes it possible to obtain a PSpace algorithm for the equational theory of binary relations.

Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K \mathbb {K} . To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K \mathbb {K} -amenability, of which we prove an algebraic characterization. It implies that normed K \mathbb {K} -amenable groups are locally elliptic, and it relates an invariant, the norm of a K \mathbb {K} -amenable group, to the order of its discrete finite p p -subquotients, where p p is the characteristic of the residue field of K \mathbb {K} . Moreover, we prove a characterization of discrete normed K \mathbb {K} -amenable groups in terms of vanishing of bounded cohomology with coefficients in K \mathbb {K} . The algebraic characterization shows that normed K \mathbb {K} -amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K \mathbb {K} coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K \mathbb {K} has positive characteristic or its residue field has characteristic 0 0 , then the comparison map is injective in all degrees. If K \mathbb {K} is a finite extension of Q p \mathbb {Q}_p , we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2 2 . A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg–Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer–Vietoris sequence, the main missing piece for computing real bounded cohomology.

Semimedial Near Rings

Objectives: To define the term ‘semimediality’ in the near ring and to demonstrate its presence with an example and to employ several attributes in near rings to analyse its features. Methods: Semimedial near rings are examined in terms of their characteristics using commutativity, distributivity and regular property. The distinction is observed using the concept of homomorphism. Findings: Every near ring which is weak commutative is proved to be semimedial and the converse exists with some additional axioms. The semimedial near ring’s anti-homomorphic image is also observed. Additionally, the correlation between the semimedial near ring and the reduced property is identified. Novelty: This study provides a novel method to medial near ring, which is the semimedial near ring. Keywords: Identity, Left Self Distributive, Nilpotent, Regular, Weak Commutative

Some additional axioms for t-normal logics. Defining K45, KB4, KD45 and S5 without using modal rules

The paper will study extensions of t-normal logics S0.5o and S0.5 using some axioms of normal logics. We will prove determination theorems for these extensions by appropriate Kripke-style models. It will allow us to obtain the determinations of the logics K45, KB4 (= KB5), KD45 and S5 without using modal rules.

Abstract embeddability ranks

We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability ranks are inspired by the nonlinear indices à la Bourgain. In particular, we resolve a problem raised by F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht in Coarse and Lipschitz universality, Fund. Math. 254 (2021), no. 2, 181–214, regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result there.

A study of rough inclusion on algebras with quasi-Boolean base

Pre-rough algebra has emerged from the rough set theory, which is a quasi-Boolean algebra with a few additional axioms. Four types of abstract rough inclusion are defined in pre-rough algebra which give rise to four different implication operators within this algebra. Properties of these four implications are studied from the angle of residuation. Logics of pre-rough algebra with respect to different implications are presented.

Cross invariance, the Shapley value, and the Shapley–Shubik power index

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Stable and weakly additive cost sharing in shortest path problems

In a shortest path problem, agents seek to ship their respective demands; and the cost on a given arc is linear in the flow. Previous works have proposed cost allocations falling in the core of the associated cooperative game. The present work combines core selection with weak versions of the additivity axiom, which allows to characterize a new family of rules. The demander rule charges each demander the cost of their shortest path, and the supplier rule charges the cost of the second-cheapest path while splitting the excess payment equally between access suppliers. With three or more agents, the demander rule is characterized by core selection and a specific version of cost additivity. Convex combinations of the demander rule and the supplier rule are axiomatized using core selection, a second version of cost additivity, and two additional axioms that ensure the fair compensation of intermediaries. With three or more agents, the demander rule is characterized by core selection and a specific version of cost additivity. Finally, convex combinations of the demander rule and the supplier rule are axiomatized using core selection, a second version of cost additivity, and two additional fairness properties.

Compositional truth with propositional tautologies and quantifier-free correctness

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as Δ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _0$$\\end{document}-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

Extensive and nonextensive statistical mechanics: Exp and log distribution functions

The dominance of Boltzmann–Gibbs distribution (BG) in statistical physics appears to endow an impression that this would be the only statistical approach available. In fact, a large class of statistical approaches already exists. This is generalizing BG statistics referring to various types of nonextensivity, nonadditivity, nonequilibrium, nonlinearity, etc. For instance, [Formula: see text] and [Formula: see text] functions play crucial roles in both extensive and nonextensive domains of statistical mechanics. Emerging in a physical system, BG statistics defines extensive entropy which relates the number of microstates to thermodynamic quantities or macroscopic states. In this regard, the Boltzmann distribution, extensive statistics, refers to a well-defined probability distribution, while the various types of nonextensive statistics categorically violate the fourth Shannon–Khinchen additivity axiom. The [Formula: see text] and [Formula: see text] distribution functions in BG, Tsallis, and generic statistics are systematically compared. We focus on the mathematical properties of both distribution functions and conclude that their compatibility exclusively depends on the nonextensive parameters, i.e. the mathematical properties of both distribution functions depend on the nonextensive parameters either that of Tsallis- or that of the generic-type of nonextensivity. We also conclude that the statistical nature of the underlying ensemble should be taken into consideration when applying the statistical approach.

An efficient solidarity value for interval cooperative games

This paper proposes an efficient interval solidarity value that operates well for interval cooperative games. In addition to the axioms of symmetry, efficiency, and additivity, this value also satisfies two new axioms, namely, interval-egalitarian A-null player and interval differential marginality. The interval-egalitarian A-null player axiom equally divides the result of the difference between the grand coalition value and the sum of the solidarity value of players in the degenerate interval game among A-null players. The interval differential marginality axiom is an interval version of the Casajus differential marginality axiom. This property states that the difference in the interval solidarity value of two players is determined by the difference between their average marginal contributions in the degenerate interval game. Eventually, the efficiency results and applicability of the proposed approach are compared with those of the other methods.

Fuzzy Metrics in Terms of Fuzzy Relations

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression d(x,y)=t. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d(x,y) and the other is a number within the range [0,+∞).

Two-level type theory and applications - ERRATUM

AbstractWe define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(\infty,1)$ -category and give some examples.

Two-level type theory and applications

AbstractWe define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(\infty,1)$ -category and give some examples.

A new axiomatization of discounted expected utility

We present a new axiomatization of the classical discounted expected utility model, which is primarily used as a decision model for consumption streams under risk. This new axiomatization characterizes discounted expected utility as a model that satisfies natural extensions of standard axioms as in the one-period case and two additional axioms. The first axiom is a weak form of time separability. It only requires that the choice between certain constant consumption streams and lotteries should be made by just taking into account the time periods where the consumption is different. The second axiom, the time–probability equivalence, requires that risk and time preferences basically work in the same way. Moreover, we prove that preferences satisfying the natural extensions of the standard axioms as well as the first axiom can be represented in a simple form relying on three functions linked to the risk or time preferences in simple situations. Finally, we illustrate that several examples that are not fully time separable satisfy all our axioms except for the time–probability equivalence.

MODULAR MANY-VALUED SEMANTICS FOR COMBINED LOGICS

AbstractWe obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics, whereas, unsurprisingly, it may be lost in the context of single-conclusion logics. Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism.

Axiomatic characterizations of the family of Weighted priority values

AbstractWe introduce a new family of values for TU‐games with a priority structure, which both contains the Priority value recently introduced by Béal et al. and the Weighted Shapley values (Kalai & Samet). Each value of this family is called a Weighted priority value and is constructed as follows. A strictly positive weight is associated with each agent and the agents are partially ordered according to a binary relation. An agent is a priority agent with respect to a coalition if it is maximal in this coalition with respect to the partial order. A Weighted priority value distributes the dividend of each coalition among the priority agents of this coalition in proportion to their weights. We provide an axiomatic characterization of the family of the Weighted Shapley values without the additivity axiom. To this end, we borrow the Priority agent out axiom from Béal et al., which is used to axiomatize the Priority value. We also reuse, in our domain, the principle of Superweak differential marginality introduced by Casajus to axiomatize the Positively weighted Shapley values. We add a new axiom of Independence of null agent position which indicates that the position of a null agent in the partial order does not affect the payoff of the other agents. Together with Efficiency, the above axioms characterize the Weighted Shapley values. We show that this axiomatic characterization holds on the subdomain where the partial order is structured by levels. This entails an alternative characterization of the Weighted Shapley values. Two alternative characterizations are obtained by replacing our principle of Superweak differential marginality by Additivity and invoking other axioms.

On Some Meta-Theoretic Topological Features of the Region Connection Calculus

This paper examines several intended topological features of the Region Connection Calculus (RCC) and argues that they are either underdetermined by the formal theory or given by the complement axiom. Conditions are identified under which the axioms of RCC are satisfied in topological models under various set restrictions. The results generalise previous results in the literature to non-strict topological models and across possible interpretations of connection. It is shown that the intended interpretation of connection and the alignment of self-connection with topological connection are underdetermined by the axioms of RCC, which suggests that additional axioms are necessary to secure these features. It is also argued that the complement axiom gives RCC models much of their topological structure. In particular, the incompatibility of RCC with interiors is argued to be given by the complement axiom.

A new model of the free monogenic digroup

It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.

A Theory of Quantum (Statistical) Measurement

We propose a theory of quantum (statistical) measurement which is close, in spirit, to Hepp’s theory, which is centered on the concepts of decoherence and macroscopic (classical) observables, and apply it to a model of the Stern-Gerlach experiment. The number N of degrees of freedom of the measuring apparatus is such that $$N \rightarrow \infty $$ , justifying the adjective “statistical”, but, in addition, and in contrast to Hepp’s approach, we make a three-fold assumption: the measurement is not instantaneous, it lasts a finite amount of time and is, up to arbitrary accuracy, performed in a finite region of space, in agreement with the additional axioms proposed by Basdevant and Dalibard. It is then shown how von Neumann’s “collapse postulate” may be avoided by a mathematically precise formulation of an argument of Gottfried, and, at the same time, Heisenberg’s “destruction of knowledge” paradox is eliminated. The fact that no irreversibility is attached to the process of measurement is shown to follow from the author’s theory of irreversibility, formulated in terms of the mean entropy, due to the latter’s property of affinity.