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Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section. This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hajek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced.

In this paper we introduce a new set of axioms that characterize uniform random dictatorship (URD) as a randomized impartial nomination rule. Unlike earlier work we use a variable population model which allows us to employ axioms that reflect consistency and proportionality–axioms that ensure that the rule behaves well with respect to the combinatorial structure of the nomination profile. Earlier work characterizing URD employed strong symmetry axioms and so it is surprising that our axioms characterize the same rule.

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory . There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach . Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L -valued rough sets with an L -set as the basic universe of defining L -valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets , we will present the axiomatic characterizations of L -valued upper and lower rough approximation operators on an L -set with respect to reflexive, symmetric, transitive L -valued relations on an L -set as well as their compositions. Choosing an L -set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

The task’s slack is the key issue to reduce the energy consumed by DVFS-enabled computing platforms. Despite the large number of scheduling algorithms that are presented in the literature, only a unique scaling axiomatic approach (SAA) is utilized in the scaling phase of the algorithms. SAA simply extends the execution of the task within its slack if a suitable scaling frequency is available. Unfortunately, when dependent-tasks applications are scheduled on such platforms, scheduling algorithms minimize the tasks’ slacks to reduce the overall completion time of the application tasks. This paper presents a mechanism that can be applied to any schedule produced by a dependent-task scheduling algorithm for both homogeneous and heterogeneous DVFS-enabled computing platforms. The proposed mechanism is called BlackLight. BlackLight attempts to extend the tasks’ slacks via rescheduling the application tasks without violating the overall completion time of the application tasks. The proposed mechanism is applied to a large number of dependent-tasks schedules of both random generated application graphs and two real-world application graphs. The experimental results based on a computer simulation show that the proposed mechanism significantly extends the tasks’ slacks compared with SAA , which leads to more reduction in the consumed energy.

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy satisfying these axioms must lie between the Renyi divergences of order 0 and $\infty $ . We further show simple conditions for positive definiteness of such relative entropies and a characterisation in terms of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

Fuzzy relations R : A 2 → [ 0 , 1 ] are widely used in decision-making in order to represent degrees of preference between alternatives in A . Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

We take the axiomatic approach to uncover the structure of the revenue-sharing problem from broadcasting sports leagues. We formalize two notions of impartiality, depending on the stance one takes with respect to the revenue generated in the games involving each pair of teams. We show that the resulting two axioms lead towards two broad categories of rules, when combined with additivity and some other basic axioms. We complement those results strengthening the impartiality notions to consider axioms of order preservation., Universidad Pablo de Olavide, Department of Economics, Carretera de Utrera, Km. 1, 41013, Seville, Spain

This paper contributes to the theory of average rate of change (ARC) measurement. We use an axiomatic approach to generalize the conventional ARC measures (such as the difference quotient and the continuously compounded growth rate) in several directions: to outcome variables with arbitrary connected domains, to not necessarily time-shift invariant dependence on time, to more general (than an interval) time sets, to a path-dependent setting, and to a benchmark-based evaluation. We also revisit and generalize the relationship between the ARC measurement and intertemporal choice models.

Hesitant fuzzy set is a natural generalization of the classical fuzzy set. A hesitant fuzzy set on a universe of discourse is in terms of a function that when applied to the universe returns a finite subset of [0, 1]. Since the axiomatic method of approximation operator is of great significance in the research of the mathematical structure of rough set theory, it is a fundamental problem in axiomatic method to find the minimum set of abstract axioms. This paper first introduces the basic concepts, properties and related operations of hesitant fuzzy set, hesitant fuzzy rough set and hesitant fuzzy rough approximation operator. Secondly, by defining inner product, outer product and by exploring their related properties, the single axiomatization problem of the classical hesitant fuzzy rough approximation operator is solved. Furthermore, we study the single axiomatization of hesitant fuzzy rough approximation operators derived from serial, reflexive, symmetric and transitive hesitant fuzzy relations, respectively. Finally, we compare and analyze the advantages and disadvantages of hesitant fuzzy set, fuzzy rough set and hesitant fuzzy rough set through some cases.

In this paper, we present constructions of Riecan, Bosbach and internal and general states on Sheffer stroke basic algebra $${\mathcal {A}}.$$ We deal with some characteristic features and independency of axiomatic system of forementioned states. We prove that a Riecan state corresponds to a Bosbach state or vice versa. In the sequel, we introduce faithful and fixed set generated by internal states on $${\mathcal {A}}.$$ We demonstrate that each Sheffer stroke basic algebra is also a Sheffer stroke basic algebra under internal state. In addition, we mention about internal state-(filter, compatible filter and prime filter) on $${\mathcal {A}}$$ and their related results. We touch on internal state congruence and investigate its relation with filters. We are interested in finding whether general states have implied Riecan and Bosbach states or not. We handle the notion of general state-morphism and some related results on $${\mathcal {A}}.$$ In order to support mentioned concepts, we give place to some attractive examples.

Purpose of the study. Analysis of the literature shows that the ordinal theory of utility is widespread in the theory of consumer behavior. To analyze consumer preferences, a utility function is used, which characterizes the value of the utility of the consumed goods and services on a scale of order. Moreover, to find the marginal utility of a product, arithmetic operations are used, which are impossible on a scale of order. To allow arithmetic operations, a quantitative analysis of the utility function is required. Consequently, the problem of quantitative measurement of the utility function is relevant.The measurement problem also arises in decision theory. For example, the hierarchy analysis method is a popular method for solving multicriteria problems, but contains an erroneous model of subjective measurement. For this reason, other methods appear in decision-making theory that should replace the method of analyzing hierarchies. The theory of the importance of criteria is being actively developed. However, the theory of the importance of criteria also does not solve the problem of quantitative measurement.For a long time, the problem of measurement has also existed in psychophysics. The existence of two mismatched psychophysical laws contradicts the classical theory of measurements. Recently, a rating solution has been proposed. The equivalence of the basic laws of psychophysics has been proved. In this paper, it is proposed to use the rating method to measure preferences in utility theory and in decision theory.Materials and methods. The domain of the rating is the set of ordered pairs of objects. Moreover, the composition (operation of addition) of objects is defined on the set of ordered pairs. A rating is a number that is assigned during a measurement to an ordered pair of objects.The rating is assumed to preserve the operation of composition of ordered pairs.An arithmetic operation is selected to carry out the measurement. The measurement result must match the result of the arithmetic operation. The result of an arithmetic operation is the difference or ratio of the values of the quantity. The rating values coincide with the result of the arithmetic operation (up to isomorphism).The additivity of the rating is used to check the adequacy of the measurement results. For this, it is assumed that the rating is independent of the measurement method. The theoretical justification for independence is the isomorphism condition. The empirical confirmation of independence is the equivalence of the basic psychophysical laws.Results. The paper presents an axiomatic approach to the measurement problem. Measurement can be carried out in both objective and subjective ways. The axiomatic and classical definition of the rating has been formulated. The axiomatic definition implies the classical definition for a special set of objects. The classic definition is constructive. To check the adequacy of the measurement results, it is enough to compare the ratings obtained by different measurement methods (method of alternatives).Conclusion. The rating method is a quantitative measurement method. The rating method can be used to construct a model of consumer behavior and in decision-making theory.

Pairwise comparisons are used in a wide variety of decision situations where the importance of alternatives should be measured on a numerical scale. One popular method to derive the priorities is based on the right eigenvector of a multiplicative pairwise comparison matrix. We consider two monotonicity axioms in this setting. First, increasing an arbitrary entry of a pairwise comparison matrix is not allowed to result in a counter-intuitive rank reversal, that is, the favoured alternative in the corresponding row cannot be ranked lower than any other alternative if this was not the case before the change (rank monotonicity). Second, the same modification should not decrease the normalised weight of the favoured alternative (weight monotonicity). Both properties are satisfied by the geometric mean method but violated by the eigenvector method. The axioms do not uniquely determine the geometric mean. The relationship between the two monotonicity properties and the Saaty inconsistency index are investigated for the eigenvector method via simulations. Even though their violation turns out not to be a usual problem even for heavily inconsistent matrices, all decision-makers should be informed about the possible occurrence of such unexpected consequences of increasing a matrix entry., Comment: 16 pages, 2 figures, 2 tables

Given the steady growth in demand for Internet resources (due to specific benefits for users) a significant issue is the governing regulation of relations that develop during the interaction of subjects of legal relations on this sphere. Intellectual property law expands the scope of regulation of legal relations via the Internet. In this regard, there is an essential need for a comprehensive study of theoretical and practical aspects of the protection of intellectual property rights on the Internet. During the study, a number of general scientific methods were used: method of observation, method of comparison, method of experiment, method of convergence from abstract to concrete, method of formalization, axiomatic method, method of analysis, and historical method. As a result of the study, the peculiarities of legal regulation of relations arising in the process of using objects of intellectual property rights on the Internet, which are caused by the difficulty in identifying the infringer of intellectual property rights, determining the place of the crime, the speed of spread of intellectual property on the Internet, low level of legal culture of the population. The ways to improve the system of protection of intellectual property rights on the Internet are the following were suggested.

Nonmodularity is a prominent property of capacity that deeply links to the internal interaction phenomenon of multiple decision criteria. Following the common architectures of the simultaneous interaction indices as well as of the bipartition interaction indices, in this paper, we construct and study the notion of probabilistic nonmodularity index and also its particular cases, such as Shapely and Banzhaf nonmodularity indices, which can be used to describe the comprehensive interaction situations of decision criteria. The connections and differences among three categories of interaction indices are also investigated and compared theoretically and empirically. It is shown that three types of interaction indices have the same roots in their first and second orders, but meanwhile the nonmodularity indices have involved less amount of subsets and can be adopted to describe the interaction phenomenon in decision analysis.

How to apply allocation in an life cycle assessment (LCA) is a long-running and controversial debate. Consensus seems to exist on the fact that the allocation procedure should follow logically from the LCA goal definition. This paper proposes to use an axiomatic method to (1) identify an allocation procedure for co-production, joint treatment, and recycling, that best responds to a specific LCA goal and (2) communicate the rationale applied by the LCA practitioner transparently. The method is illustrated via a case study. The specific goal definition for which a suitable allocation procedure is identified is to evaluate what impacts can be attributed to a product, which could inform a company about potential sources of reputation damage. Subjective assumptions that reflect our vision of “what impacts can be attributed to a product” are described in definitions and axioms. Axioms are formulated that describe the system boundaries of the product system and the partitioning criterion. The derived allocation procedure corresponds to “Allocation at the Point of Substitution,” which is applied in one of the system models of ecoinvent. Partitioning is based on market information, which corresponds to a cause-oriented perspective on “what impacts can be attributed to a product.” Other LCA goal definitions and rationales could require different system boundaries or a different partitioning criterion. The axiomatic method presented in this paper supports the identification of a suitable allocation procedure for a defined LCA goal and the transparent communication of the rationale that backs up this procedure. Building forth on the approach of this paper, a collection of axioms and corresponding allocation procedures could be developed on which consensus might exist within the LCA community. Such goal-dependent allocation procedures could form the basis of future guidance on LCA.