摘要 : There exist several versions of discrete counterpart of continuity in the framework of finite chains, e.g., the smoothness, the divisibility, intermediate-value property and the 1-Lipschitz property. In this paper, we first discuss the relationships among the smoothness, divisibility, intermediate-value property and 1-Lipschitz property. Second, we present complete characterizations of divisible associative aggregation operations on finite chains.

摘要 : This study presents characterizations of three classes of idempotent uninorms on a bounded lattice by the orders of their associated meet-semilattices. The first one is the class of internal uninorms, the second one is the class of idempotent uninorms defined on a lattice in which all elements are comparable with the corresponding neutral element and the third one is the class of idempotent uninorms defined on a lattice in which a single point is incomparablewith the corresponding neutral element.

摘要 : In this study, a new method, called g-sum, is presented to join two posets together that generalizes the linear sum oftwo posets. Idempotent uninorms on a bounded chain are in one-to-one correspondence with special linear orders on itand g-sum can be used to construct such special linear orders.

摘要 : This paper focuses on the topic of ordinal sums of semigroups in the sense of A. H. Clifford - a method for constructing a new semigroup from a given system of semigroups indexed by a linearly ordered index set. We completely describe the linearly ordered index set for an ordinal sum of semigroups yielding a uninorm.

摘要 : Ouyang et al. characterized idempotent uninorms on a complete chain in terms of decreasing unary functions with a symmetry-related property. In this paper, we generalize the results of Ouyang et al. and show that each idempotent uninorm on a bounded chain can be extended to an idempotent uninorm on a complete chain isomorphic to its Dedekind-MacNeille completion such that the restriction of the latter idempotent uninorm to this bounded chain coincides with the previous uninorm, which, together with the results of Ouyang et al. [7], gives a characterization of idempotent uninorms on a bounded chain. © 2023 Elsevier B.V.

摘要 : A connected order topological space, which is much simper than a real interval, is considered for the assignment problem of a rational aggregation by Fung and Fu, and they described the structure of an idempotent, commutative, associative, increasing and continuous binary operation on such a space. Fodor extended Fung-Fu's result to the non-commutative case. The aim of the present paper is to generalize Fodor's result to the non-idempotent case. © 2023 Elsevier B.V.

摘要 : Conditional distributivity (also called restricted distributivity) is a form of relaxed distributivity on the restricted domain. There are three options for conditional distributivity in literature. In this paper, we focus on type-III conditional distributivity equation involving nullnorms over uninorms. Firstly, we show that it can be transformed into the other two types of conditional distributivity equations involving t-norms/t-conorms over uninorms. Secondly, type-I and type-II conditional distributivity equations involving t-norms/t-conorms over uninorms that are related to our study and remain unknown are investigated. Finally, we show that this new perspective results in not only all known solutions, but new solutions as well.

摘要 : We embed the Aubry set and Mather set in the Tonelli Hamiltonian system to the contact Hamiltonian system. We find the embedded Aubry set is the set of non-wandering points of the contact Hamiltonian system and the Mather set is the support of set of the invariant Borel probability measures for the contact Euler—Lagrange flow. From this viewpoint, we can conclude that Aubry set and Mather set are symplectic invariants.

摘要 : This paper is a contribution to the topic of ordinal sums of semigroups in the sense of A. H. Clifford. The goal of this article is twofold. Firstly, we present sufficient and necessary conditions for obtaining uninorms as an ordinal sum of semigroups, of which the underlying sets are non-empty subintervals of the unit interval [0, 1]. Secondly, we show that each uninorm locally internal on A(e) can be decomposed into an ordinal sum of such semigroups.

摘要 : Homogeneity, which plays an essential role in decision making, economics and image processing, reflects the regularity of aggregation functions with respect to the inputs with the same ratio. Quasi-homogeneity is a relaxed homogeneity that reflects the original output as well as the same ratio. This paper is devoted to the characterization of all homogeneous/quasi-homogeneous binary aggregation functions in terms of single-argument functions.