Horizontal Sum Research Articles

Commutative, associative and monotone functions on horizontal sum of chains

The commutative, associative and monotone (CAM) functions on a horizontal sum of bounded chains are studied. Depending on their range, these functions are divided into two main classes and properties of CAM functions from each of these classes are described. Several necessary and sufficient conditions under which a CAM function defined on a horizontal sum of bounded chains can be expressed as a non-trivial (z)-ordinal sum of semigroups are also introduced. Special classes of CAM functions are discussed and several examples are included. Various construction methods for special CAM functions defined on a horizontal sum of bounded chains, such as t-norms, t-conorms, uninorms and nullnorms, are also presented.

Detecting slow eye movements with bimodal-LSTM for recognizing drivers’ sleep onset period

Slow eye movements (SEMs) indicate sleep onset period but are rarely studied in the field of driver fatigue detection. Through visual observation and statistical analysis we find that SEMs occur during eye-closed periods and have a high overall coverage rate during driving, which makes it feasible to utilize detecting SEMs to recognize drivers’ sleep onset period. To detect SEMs, we adopt a bimodal Long Short-Term Memory (LSTM) network to deal with temporal information and multimodal information in physiological signals. To extend the distinguishing information, we define a novel horizontal sum (HSUM) signal, which is sum of signals from two horizontal electrooculogram (EOG) channels. Electroencephalogram (EEG)-related features are extracted from the HSUM signal in contrast to those from the traditional O2 signal. EOG features are extracted from the horizontal EOG (HEOG) signals. The results demonstrate that features from multimodal signals (HSUM and HEOG signals, or O2 and HEOG signals) achieve better classification results than features from the single HEOG signal. And the EEG-related features extracted from self-defined HSUM signals achieve comparable results to those from traditional O2 signals, thus avoiding using extra O2 channel. The proposed method of detecting SEMs using the bimodal LSTM to classify features from HSUM and HEOG signals achieves the average F-score of 76.5%, which is higher than the classic support vector machine by 7.5%. This method of detecting SEMs using only two channels helps to build a user acceptable and feasible system for recognizing drivers’ sleep onset period.

Natural partial order induced by a commutative, associative and idempotent function

We study the natural partial order of commutative, associative, idempotent functions, covering as a special case t-norms, t-conorms, uninorms and nullnorms on bounded lattices. Unlike other approaches known in the literature, where the order induced by a given function is defined with respect to the original order on the given bounded lattice, we use original ideas of Hartwig, Nambooripad and Mitsch and define the natural partial order on an arbitrary set without connection to any order. We show that the natural partial order induced by any commutative, associative, idempotent function F corresponds to a meet semi-lattice and that F(x,y) can be expressed by the meet of x and y with respect to the natural partial order. This result can be used for an easy verification of the associativity of a commutative, idempotent function. Further, in some special cases we show the necessary and the sufficient conditions for such a function to be non-decreasing in each coordinate. We show that the natural meet semi-lattice connects the ordinal sum in the sense of Clifford and the ordinal sum in the sense of Birkhoff. An example of an ordinal sum of functions (which need not to be idempotent) on a horizontal sum lattice is also introduced.

Lattice-based sum of t-norms on bounded lattices

The concept of ordinal sums in the sense of Clifford have long been blamed for their limitations in constructing new t-norms including inability to cope with general bounded lattices. Motivated by this observation, and based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al., we propose a new sum-type construction of t-norms, called a lattice-based sum of t-norms, for building new t-norms on bounded lattices from given ones. The proposed sum is generalizing the well-known ordinal and horizontal sum constructions of t-norms by allowing for lattice ordered index sets. We demonstrate that, like the ordinal sum of t-norms, the lattice-based sum of t-norms can be generalized using as summands so-called t-subnorms, still leading to a t-norm. Subsequently, we apply the results for constructing several new families of t-norms and t-subnorms on bounded lattices. In the same spirit, by the duality, we will also introduce lattice-based sums of t-conorms and t-subconorms.

Assessment of objective and subjective measures as indicators for facial esthetics.

Background:The study mentioned was aimed to examine the contribution of the objective measures representing anterior-posterior (AP) and vertical characteristics, dental esthetics, or their combination that are used in daily orthodontic practice in the assessment of the facial esthetics.Materials and Methods:A panel of 64 laypersons evaluated the facial esthetics of 32 boys and 32 girls, stratified over four different angle classes, on a visual analog scale. The relationship between the objective parameters and facial esthetics was evaluated by the backward multiple regression analysis.Results:Dental esthetics, expressed by the esthetic component of the index of orthodontic treatment need (AC/IOTN), appeared to be the most vital indicator for facial esthetics. The horizontal sum, a variable for AP characteristics of the patient, could be a better variable when compared with the overjet.Conclusion:Addition of this newly defined parameter to the AC/IOTN improved the prognostic value from 25% to 35%.

Lattice-based sums

In this contribution, the well-known ordinal sum technique of posets is generalized by allowing for a lattice ordered index set instead of a linearly ordered index set, and we argue for the merits of this generalization. We will call such a proposed sum-type construction a lattice-based sum. Our new approach of lattice-based sum extends also the horizontal sum. We show that the lattice-based sum of posets is again a poset. Subsequently, we apply the results for constructing new lattices by investigating lattice-based sums when the summand posets are lattices. We show that under certain assumptions, the lattice-based sum of lattices will be a lattice.

The Variety of Lattice Effect Algebras Generated by MV-algebras and the Horizontal Sum of Two 3-element Chains

It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum $${\mathcal{H}}$$ of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and $${\mathcal{H}}$$ is definable by three simple identities and the problem now is to check if these identities are satisfied by all distributive lattice effect algebras or not.

On extensions of triangular norms on bounded lattices

Smallest and largest possible extensions of triangular norms on bounded lattices are discussed. As such ordinal and horizontal sum like constructions for t-norms on bounded lattices are investigated. Necessary and sufficient conditions for the lattice guaranteeing that the extension is again a t-norm are revealed.

Objective measures as indicators for facial esthetics in white adolescents.

The objective of this study was to examine the contribution of objective measures representing anterior-posterior and vertical characteristics, dental esthetics, or their combination that are used in daily orthodontic practice in the assessment of facial esthetics. A panel of 78 laymen evaluated facial esthetics of 32 boys and 32 girls, stratified over the four Angle classes, on a visual analogue scale. The relation between the objective parameters and facial esthetics was evaluated by backward multiple regression analysis. Dental esthetics as expressed by the Aesthetic Component of the Index of Orthodontic Treatment Need (AC/IOTN) appeared to be the most important indicator for facial esthetics. A new parameter, the "horizontal sum" was found to be a reliable variable for the anterior-posterior characteristics of the patient. Addition of this newly defined parameter to the AC/IOTN improved the prognostic value from 25% to 31%.

Intervals in lattices of quasiorders

We investigate the structure of intervals in the lattice of all closed quasiorders on a compact or discrete space. As a first step, we show that if the intervalI has no infinite chains then the underlying space may be assumed to be finite, and in particular,I must be finite, too. We compute several upper bounds for its size in terms of its heighth, which in turn can be computed easily by means of the least and the greatest element ofI. The cover degreec of the interval (i.e. the maximal number of atoms in a subinterval) is less than 4h. Moreover, ifc⩾4(n−1) thenI contains a Boolean subinterval of size 2 n , and ifI is geometric then it is already a finite Boolean lattice. While every finite distributive lattice is isomorphic to some interval of quasiorders, we show that a nondistributive finite interval of quasiorders is neither a vertical sum nor a horizontal sum of two lattices, with exception of the pentagon. Many further lattices are excluded from the class of intervals of quasiorders by the fact that no join-irreducible element of such an interval can have two incomparable join-irreducible complements. Up to isomorphism, we determine all quasiorder intervals with less than 9 elements and all quasiorder intervals with two complementary atoms or coatoms.

Bell-type inequalities in horizontal sums of Boolean algebras

We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.

Combined systems in quantum probability

A framework for quantum probability is developed and combinations of systems are studied within this framework. In particular, we consider the horizontal sum, the direct sum, and the Cartesian product of quantum probability systems. The relations between these combinations and the concepts of interference and independence of measurements are derived. We also consider the amplitude superselection structure of these combinations.

Varieties of Orthomodular Lattices

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],