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Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system—which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit. [ABSTRACT FROM AUTHOR]

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called ⪯-factorizations, for the ⪯-non-units of a (multiplicatively written) monoid H endowed with a preorder ⪯, where an element u ∈ H is a ⪯-unit if u ⪯ 1 H ⪯ u and a ⪯-non-unit otherwise. The "building blocks" of these factorizations are the ⪯-irreducibles of H (i.e., the ⪯-non-units a ∈ H that cannot be written as a product of two ⪯-non-units each of which is strictly ⪯-smaller than a); and it is interesting to look for sufficient conditions for the ⪯-factorizations of a ⪯-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal ⪯-factorizations (i.e., a refinement of ⪯-factorizations used to counter the "blow-up phenomena" that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the "local behavior" of the pair (H , ⪯). Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where ⪯ is the divisibility preorder on H (and hence in the setup of the classical theory). [ABSTRACT FROM AUTHOR]

Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one. [ABSTRACT FROM AUTHOR]

A quasiconvex function f being given, does there exist an increasing and continuous function k which makes k ∘ f convex? How to build such a k? Some words on least convex (concave) functions. The ratio of two positive numbers is neither locally convexifiable nor locally concavifiable. Finally, some considerations on the approximation of a preorder from a finite number of observations and on the revealed preference problem are discussed. [ABSTRACT FROM AUTHOR]

We study the NFA reductions by invariant equivalences and preorders. It is well-known that the NFA minimization problem is PSPACE-complete. Therefore, there have been many approaches to reduce the size of NFAs in low polynomial time by computing invariant equivalence or preorder relation and merging the states within same equivalence class. Here we consider the nondeterminism reduction of NFAs by invariant equivalences and preorders. We, in particular, show that computing equivalence and preorder relation from the left is more useful than the right for reducing the degree of nondeterminism in NFAs. We also present experimental evidence for showing that NFA reduction from the left achieves the better reduction of nondeterminism than reduction from the right. [ABSTRACT FROM AUTHOR]

We first take a whimsical look at a couple of questions relating to categories as generalized posets. Then we study the question of functorial choice of pullbacks. Finally, we consider a simple question in basic category theory, with an elementary solution which is surprisingly difficult to generalize to 2-categories. [ABSTRACT FROM AUTHOR]

A topological preordered space is a topological space with a preorder ⪯, that is, a binary relation which is reflexive and transitive. Nachbin generalized Tietze's theorem and Urysohn's theorem for continuous increasing functions on a normally preordered space, where a real-valued function f is called increasing if f (x) ≤ f (x ′) whenever x ⪯ x ′. In this paper, we provide an increasing version of Gillman-Jerison theorem characterizing C ⁎ -embedding. Namely, we show that every bounded continuous increasing function on a subspace A of a topological preordered space X can be extended over X if and only if every pair of completely order separated subsets of A can be completely order separated in X. Various theorems and several counterexamples of extensions of continuous increasing functions are also given. [ABSTRACT FROM AUTHOR]

In this work, we will introduce the notion of i-continuity and d-continuity of the functions between the ordered closure spaces. Then, we give a classification of the closure operators defined on the set of i-continuous and d-continuous functions. [ABSTRACT FROM AUTHOR]

In this work, we focus on adjunctions, also called isotone Galois connections, in the framework of fuzzy preordered sets (hereafter, fuzzy preposets). Specifically, we present necessary and sufficient conditions so that, given a mapping f : A → B from a fuzzy preposet A into an unstructured set B , it is possible to construct a suitable fuzzy preorder relation on B for which there exists a mapping g : B → A such that the pair ( f , g ) constitutes an adjunction. [ABSTRACT FROM AUTHOR]

The rough intuitionistic fuzzy set theory is an extension of the theory of rough fuzzy sets. For further studying the theories and applications of rough intuitionistic fuzzy sets, in this paper, we propose a type of rough intuitionistic fuzzy sets and investigate its topological structure. It is proved that an intuitionistic fuzzy topology is induced by a binary relation in a crisp approximation space, and a preorder is generated by a family of intuitionistic fuzzy sets. Moreover, there exists a one-to-one correspondence between the set of all intuitionistic fuzzy topologies having property (*) and the set of all preorders. That is to say, there exists a one-to-one correspondence between the set of all intuitionistic fuzzy topological spaces having property (*) and the set of all crisp approximation spaces whose relations are preorders. [ABSTRACT FROM AUTHOR]

The rough intuitionistic fuzzy set theory is an extension of the theory of rough fuzzy sets. For further studying the theories and applications of rough intuitionistic fuzzy sets, in this paper, we propose a type of rough intuitionistic fuzzy sets and investigate its topological structure. It is proved that an intuitionistic fuzzy topology is induced by a binary relation in a crisp approximation space, and a preorder is generated by a family of intuitionistic fuzzy sets. Moreover, there exists a one-to-one correspondence between the set of all intuitionistic fuzzy topologies having property (*) and the set of all preorders. That is to say, there exists a one-to-one correspondence between the set of all intuitionistic fuzzy topological spaces having property (*) and the set of all crisp approximation spaces whose relations are preorders. [ABSTRACT FROM AUTHOR]

The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called Cartesian and is studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma I, are called I-Cartesian and are characterized. The characterization reveals a hidden structure on such spaces. Several other characterizations are obtained, and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well. [ABSTRACT FROM AUTHOR]

The number of topologies (non-homeomorphic topologies) on a fixed finite set having n elements are now known up to n = 18 ( n = 16 respectively) but still no complete formula yet. There are one to one correspondences among topologies, preorders and transitive digraphs on a given finite set. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite simple graph. [ABSTRACT FROM AUTHOR]

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, motivated by results in [S.S. Dragomir, Quasilinearity of some functionals associated with monotonic convex functions, RGMIA Res. Rep. Coll. 14 (2011), Artcle 24, 13 pp.] and [M. Niezgoda, Cone orderings, group majorizations and similarly separable sequences, Linear Algebra Appl. 436 (3) (2012) pp. 579–594], we investigate maps involving quotient with an -subadditive operator Φ and a positive superadditive functional φ. We pay special attention to vector-valued operators Φ. We study the subadditivity of the map with (-increasing) convex f. Next, we give applications for group majorization preorders. We establish some preorder-preservation results for the map with a group majorization and a combination of preorders and . In consequence, we show that the function is -increasing. [Copyright &y& Elsevier]

Abstract: Generalizations of the classic Grüss inequality in normed spaces are given. The results are illustrated with some specific examples. [Copyright &y& Elsevier]

The Dempster-Shafer theory of belief functions has proved to be a powerful formalism for uncertain reasoning. However, belief functions on a finite frame of discernment Ω are usually defined in the power set 2, resulting in exponential complexity of the operations involved in this framework, such as combination rules. When Ω is linearly ordered, a usual trick is to work only with intervals, which drastically reduces the complexity of calculations. In this paper, we show that this trick can be extrapolated to frames endowed with an arbitrary lattice structure, not necessarily a linear order. This principle makes it possible to apply the Dempster-Shafer framework to very large frames such as the power set, the set of partitions, or the set of preorders of a finite set. Applications to multi-label classification, ensemble clustering and preference aggregation are demonstrated. [ABSTRACT FROM AUTHOR]

It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them all via filters. To achieve this, we introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). We study congruences which can be described via filters. We show that the lattice of semilattice (left normal band, right normal band) congruences is a homomorphic image of the lattice of negative (right negative, left negative) operation-compatible quasiorders. [ABSTRACT FROM AUTHOR]

In the context of crisp or classical relations, one may find several alternative characterizations of the concept of a total preorder. In this contribution, we first discuss the way of translating those characterizations to the framework of fuzzy relations. Those new properties depend on t-norms. We focus on two important families of t-norms, namely those that do not admit zero divisors and those that are rotation invariant. For these families, we study whether or not the properties shown for fuzzy relations lead to characterizations of complete fuzzy preorders. Special attention is paid to the minimum operator, which shows the best behaviour in preserving most of the characterizations known for crisp relations. [ABSTRACT FROM AUTHOR]

In this paper, we propose an index to measure the agreement level of an individual preorder with respect to a collective preorder (or reference preorder). The originality of the proposed index consists in the two following facts: (i) it quantifies the agreement level between two preorders which are not necessarily complete (i.e. they may include the incomparability relation), a preference system rarely tackled in the literature and (ii) it takes into account the positions of the alternatives in the individual preorder. The proposed index is then used to compute, within the framework of a consensus preorder research, the agreement level of all group's members with respect to a collective preorder. [ABSTRACT FROM AUTHOR]

Abstract: Removing the condition of symmetry in the notion of a fuzzy (pseudo)metric, in Kramosil and Michalek''s sense, one has the notion of a fuzzy quasi-(pseudo-)metric. Then for each fuzzy quasi-pseudo-metric on a set X we construct a fuzzy quasi-pseudo-metric on the collection of all nonempty subsets of X, called the Hausdorff fuzzy quasi-pseudo-metric. We investigate several properties of this structure and present several illustrative examples as well as an application to the domain of words. The notion of Hausdorff fuzzy quasi-pseudo-metric when quasi-pseudo-metric fuzziness is considered in the sense of George and Veeramani is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper we study translation-invariant maps on a linear space. A method for estimating quantities induced by such maps is presented. Applications for differences of operators, commutators, G-contractive maps and Grüss type inequalities are also given. [Copyright &y& Elsevier]

This article studies multi-class loss systems with two layers of servers, where each server at the first layer is dedicated to a certain customer class, while the servers at the second layer can handle all customer classes. The routing of customers follows an overflow scheme, where arriving customers are preferentially directed to the first layer. Stochastic comparison and coupling techniques are developed for studying how the system is affected by the packing of customers, altered service rates, and altered server configurations. This analysis leads to computationally fast upper and lower bounds for the performance of the system. [ABSTRACT FROM AUTHOR]

In this paper, we present an Ekeland type variational principle, Takahashi type minimization theorem, Caristi type fixed point theorem and a generalized Oettli-Thera's equivalent form on complete D-metric space. We also prove that they are equivalent. The results present in this paper unify, improve and extend many known results due to Zhong, Ekeland, Takahashi, Caristi, and others. [ABSTRACT FROM AUTHOR]

Abstract: We give a generalization of Laguerre–Samuelson’s inequality in the framework of group theory. This extends a recent result of Trenkler and Puntanen [G. Trenkler, S. Puntanen, A multivariate version of Samuelson’s inequality, Linear Algebra Appl. 410 (2005) 143–149]. Some examples are given. [Copyright &y& Elsevier]

The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output (MIMO) coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotonic in D†D in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices! This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels (MAC) are also provided. [ABSTRACT FROM AUTHOR]

Let P be a preorder (i.e., reflexive, transitive relation) on a finite set X. The ideal polynomial of P is the function $${\text{ideal}}_{P} (x) = {\sum\nolimits_{k \geqslant 0} {d_{k} x^{k} ,} }$$ where d k is the number of ideals (i.e. downwards closed sets) of cardinality k in P. We provide upper bounds for the moduli of the roots of ideal P( x) in terms of the width of P. We also provide examples of preorders with roots of large moduli. The results have direct applications to the generating polynomials counting open sets in finite topologies. [ABSTRACT FROM AUTHOR]

Abstract: We present different techniques for reducing the number of states and transitions in nondeterministic automata. These techniques are based on the two preorders over the set of states, related to the inclusion of left and right languages. Since their exact computation is -hard, we focus on polynomial approximations which enable a reduction of the NFA all the same. Our main algorithm relies on a first approximation, which can be easily implemented by means of matrix products with an time complexity, and optimized to an time complexity, where m is the number of transitions and n is the number of states. This first algorithm appears to be more efficient than the known techniques based on equivalence relations as described by Lucian Ilie and Sheng Yu. Afterwards, we briefly describe some more accurate approximations and the exact (but exponential) calculation of these preorders by means of determinization. [Copyright &y& Elsevier]

In this paper we compare four notions of fair simulation: direct , delay , game , and exists . Our comparison refers to three main aspects: The time complexity of constructing the fair simulation, the ability to use it for minimization, and the relationship between the fair simulations and universal branching-time logics. We developed a practical application that is based on this comparison. The application is a new implementation for the assume-guarantee modular framework presented By Grumberg at al. in [ACM Transactions on Programming Languages and Systems (TOPLAS), 16 (1994) 843]. The new implementation significantly improves the complexity of the framework. [Copyright &y& Elsevier]

In the algebra of single-valued structures, cyclicity is one of the fundamental properties of groups. Therefore, it is natural to study it also in the algebra of multivalued structures (algebraic hyperstructure theory). However, when one considers the nature of generalizing this property, at least two (or rather three) approaches seem natural. Historically, all of these had been introduced and studied by 1990. However, since most of the results had originally been published in journals without proper international impact and later—without the possibility to include proper background and context-synthetized in books, the current way of treating the concept of cyclicity in the algebraic hyperstructure theory is often rather confusing. Therefore, we start our paper with a rather long introduction giving an overview and motivation of existing approaches to the cyclicity in algebraic hyperstructures. In the second part of our paper, we relate these to E L -hyperstructures, a broad class of algebraic hyperstructures constructed from (pre)ordered (semi)groups, which were defined and started to be studied much later than sources discussed in the introduction were published. [ABSTRACT FROM AUTHOR]