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An AG-groupoid also called a Left-almost semigroup is a magma satisfying the law, uv • w = wv • u Vu, v, w. In this paper, the concept of Cheban AG-groupoid is developed and investigated as a subclass of AG-groupoid. Various non-associative examples and counterexamples are constructed by the recent computational techniques of Mace-4, GAP and Prover-9. Cheban AG-groupoids are enumerated up to order 6 and various relations of this class are established with already existing subclasses of AG-groupoids. Furthermore, ideals in Cheban AG-groupoid are introduced and investigated. [ABSTRACT FROM AUTHOR]

We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fullybracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, right rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. We then describe two main applications of the coherence theorem, including: 1. A new proof of the lattice property for the Tamari order, and 2. A new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Yn. [ABSTRACT FROM AUTHOR]

The D numbers theory is a novel theory to express uncertain information. It successfully overcomes some shortcomings of Dempster-Shafer theory, such as the conditions of exclusiveness hypothesis and completeness constraint. However, the combination rule of D numbers does not satisfy the associative property, which leads to limitations in practical application for D numbers. In this paper, the improved D numbers theory is proposed to overcome the weakness based on the analysis of D numbers’ combination rule. A new algorithm is constructed with the strict proof to simplify the combination rule. The similarities and differences among DS theory, D numbers, and the improved D numbers are introduced with the numerical analysis. An illustrative example of the radiation source identification is presented to demonstrate the effectiveness of the improved method. [ABSTRACT FROM AUTHOR]

An ordered AG-groupoid can be referred to as an ordered left almost semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest one-sided ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and fuzzy one-sided ideals. We also introduce the concept of an ordered AG⁎⁎⁎-groupoid and investigate its structural properties by using the generated ideals and fuzzy one-sided ideals. These concepts will verify the existing characterizations and will help in achieving more generalized results in future works. [ABSTRACT FROM AUTHOR]

Associative classification (AC) integrates the task of mining association rules with the classification task to increase the efficiency of the classification process. AC algorithms produce accurate classification and generate easy to understand rules. However, AC algorithms suffer from two drawbacks: the large number of classification rules, and using different pruning methods that may remove vital information to achieve the right decision. In this paper, a new hybrid AC algorithm (HAC) is proposed. HAC applies the power of the Naïve Bayes (NB) algorithm to reduce the number of classification rules and to produce several rules that represent each attribute value. Two experiments are conducted on an Arabic textual dataset and the standard Reuters-21578 datasets using six different algorithms, namely J48, NB, classification based on associations (CBA), multi-class classification based on association rules (MCAR), expert multi-class classification based on association rules (EMCAR), and fast associative classification algorithm (FACA). The results of the experiments showed that the HAC approach produced higher classification accuracy than MCAR, CBA, EMCAR, FACA, J48 and NB with gains of 3.95%, 6.58%, 3.48%, 1.18%, 5.37% and 8.05% respectively. Furthermore, on Reuters-21578 datasets, the results indicated that the HAC algorithm has an excellent and stable performance in terms of classification accuracy and F measure. [ABSTRACT FROM AUTHOR]

Left Bol loops with the A ℓ -property or gyrogroups are generalization of groups which do not explicitly have associativity. In this work, we define topological gyrogroups and study some properties of them. In spite of having a weaker algebraic form, topological gyrogroups carry almost the same basic properties owned by topological groups. In particular, we prove that being T 0 and T 3 are equivalent in topological gyrogroups. Furthermore, a topological gyrogroup is first countable if and only if it is premetrizable. Finally, a direct product of topological gyrogroups is a topological gyrogroup. [ABSTRACT FROM AUTHOR]

This paper introduces a class of pseudo-orbits which guarantees the same lower bound error (LBE) for two different natural interval extensions of discrete maps. In previous work, the LBE was investigated along with a simple technique to evaluate numerical accuracy of free-run simulations of polynomial NARMAX or similar discrete maps. Here we prove that it is possible to calculate the LBE for two pseudo-orbits, extending so the results of previous work in which the LBE is valid for only one of the two pseudo-orbits. The main application of this technique is to provide a simple estimation of the LBE. We illustrate our approach with the Logistic Map and Hénon Map. Using double precision, our results show that we ought simulate the Logistic Map and Hénon Map with less than 100 iterations, which is, for instance, far less than the number usually considered as transient to build bifurcation diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if $$(X,*)$$ is a groupoid and if $$\varphi : X^2\rightarrow X^2$$ is a function $$\varphi (a, b) = (u, v)$$ , then $$(X,*)$$ is a left-twisted semigroup with respect to $$\varphi $$ if for all $$a, b, c\in X$$ , $$a*(b*c) = (u*v)*c$$ . Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule $$(a*b)*c = u*(v*c)$$ for all $$a, b, c\in X$$ . Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field $$(X,+, \cdot )$$ via a formula $$x*y=\lambda x + \mu y$$ , with $$\lambda , \mu \in X$$ , fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of $$(X,*)$$ implies the fact that $$(X,*)$$ is a semigroup. [ABSTRACT FROM AUTHOR]

The problem of distributivity was posed many years ago and it has been investigated for families of certain operations, such as t-norms, t-conorms, uninorms and nullnorms. In this paper, we continue to investigate the same topic as the above by focusing on semi-nullnorms and semi-t-operators, which are generalization of nullnorms and t-operators by omitting commutativity and associativity, and commutativity, respectively. The obtained results are not only the full characterization of this kind of distributivity equation, but also extensions of distributivity for nullnorms and for semi-nullnorms. Moreover, this paper finishes the full characterization of distributivity between semi-nullnorms and semi-t-operators together with Refs. 6 and 20. [ABSTRACT FROM AUTHOR]

We study so-called near semirings endowed with an antitone involution. Such a near semiring is in fact a bounded lattice which has one more binary operation, the multiplication. We classify several families of bounded lattices which can be organized in such near semirings, e.g. chains or orthomodular lattices. A particular case are the so-called balanced near semirings which form a variety which is congruence distributive, permutable and regular. [ABSTRACT FROM AUTHOR]

Background: Specialized content knowledge (SCK) is a type of mathematical content knowledge specifically needed for teaching. This type of knowledge, although serving as a critical component for preservice teacher education, is often challenging to develop with preservice elementary teachers (PTs). The purpose of this study is to investigate PTs' development of SCK for teaching fundamental mathematical ideas in a university methods course. Focusing on the case of the associative property (AP) of multiplication, the author as the course instructor identified three instructional opportunities (a formal introduction and two delayed revisits) to stress two SCK components, representations, and explanations. PTs' learning progresses were assessed through three diagnostic tests (a pretest, a mid-term exam, and a final exam) and two prompts, which informed the upcoming lesson design. Meanwhile, the course instructor conducted ongoing reflections on PTs' learning, which also informed the corresponding lessons. Results: It was found that PTs initially generated abstract number sentences without reasoning about the contexts of word problems. This representational sequence indicates a symbol precedence view. When prompted for explanations, PTs focused on individual numbers rather than quantitative relationships, and they could not consistently apply the basic meaning of multiplication for reasoning. The methods course, when designed to address these issues, promoted PTs' SCK development. At the end of the course, the majority of PTs were able to generate number sentences based on the word problem structures and provided reasonable explanations; however, the methods course also faced dilemmas due to PTs' robust symbol precedence view and the tension between PTs' learning and children's learning. Conclusions: Very few studies have explored ways to support PTs' knowledge growth in SCK, especially for teaching fundamental mathematical ideas. This study, by carefully documenting the successes and challenges in developing PTs' SCK, contributes to the existing literature. Based on our findings, this study highlights the importance of stressing basic meanings so as to develop PTs' explanation skills. Meanwhile, to develop PTs' representation skills, university instructors should be aware of the tension between PTs' and children's learning as manifested by PTs' symbol precedence view. Finally, to support PTs' SCK growth, it is also important to emphasize the role of elementary textbooks. [ABSTRACT FROM AUTHOR]

In this paper, a new product of tensors is given, which is a generalization of the product of tensors introduced by Gelfand, Kapranov, Zelevinsky. This product satisfies the associative law. Using the associative law, it is concluded that the product of tensors given by Bu and others and the general product of tensors introduced by Shao can be represented by this new tensors product. As an application of this new tensors product and the associative law, the hyperdeterminant of the product of a kind of tensors is studied, i.e., some formulas for the hyperdeterminant of boundary formats product are shown. [ABSTRACT FROM AUTHOR]

We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the compositions are strictly associative and unitary, whereas in the other two, these laws only hold up to coherent isomorphism. The main feature is that the interchange law between the second and third compositions does not hold, but rather there is a non-invertible comparison cell which satisfies some coherence conditions. We introduce appropriate morphisms of intercategory, of which there are three types, and cells relating these. We show that these fit together to produce a strict triple category of intercategories. [ABSTRACT FROM AUTHOR]

In this paper we present a novel associative classification algorithm for visual concept detection using interest points (color descriptors). Our algorithm relies on fuzzy (soft) partitions to generate a per-descriptor record model which is used for training associative classifiers. In this paper, we use the first 10000 images of the MIR Flickr dataset (with 38 visual concepts -- 14 regular annotation topics and 24 pre-annotations) for evaluating our algorithm. Because of its fuzzy nature, our approach also handles polysemy and synonymy (common problems in most crisp (non-fuzzy) image classifiers) very well. As associative classification leverages frequent patterns mined from a given dataset, its performance as adjudged from average area under curve (AUC) values over three-fold cross validation is very good, especially for regular annotation topics and for "scenic" concepts. Our approach has the added advantage that the rules used for classification have clear semantics, and can be comprehended easily, unlike other classifiers, such as SVM, which act as black-boxes. From an empirical perspective (on the MIR Flickr dataset), the performance of our algorithm is much better, than that of either bagof- words (Linear SVM and SVM RBF Kernel) or Linear SVM based on fuzzy (soft) partitions -- both benchmark approaches are based on interest points (color descriptors). [ABSTRACT FROM AUTHOR]

Abstract: We study a general product of two n-dimensional tensors and with orders and . This product satisfies the associative law, and is a generalization of the usual matrix product. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors. [Copyright &y& Elsevier]

We investigate nonsingular hypercubes, and prove several results, stating a condition for a hypercube to be the product of hypercubes of smaller dimensions. There is a shortage of higher dimensional nonsingular hypercubes in the literature. However, we show that the product of two nonsingular hypercubes is always nonsingular. Then we show how to construct four-dimensional nonsingular hypercubes that are not the products of two three-dimensional hypercubes. It is noted that higher dimensional nonsingular hypercubes, that are not products of smaller ones, correspond to many semifields. [ABSTRACT FROM AUTHOR]

We prove that an cancellative n-groupoidAcan be homotopic embedded in an n-group if only if in A are satisfied all n-ary Malcev conditions. Now we shall prove that in the presence of associative law we obtain homomorphic embeddings. Furthermore if A has a lateral identity a such embeddings is assured by a subset of n-ary Malcev conditions - unary Malcev conditions. [ABSTRACT FROM AUTHOR]

This paper describes the ecosystem of R add-on packages developed around the infrastructure provided by the package arules. The packages provide comprehensive functionality for analyzing interesting patterns including frequent itemsets, association rules, frequent sequences and for building applications like associative classification. After discussing the ecosystem's design we illustrate the ease of mining and visualizing rules with a short example. [ABSTRACT FROM AUTHOR]

We define algebras of triple semigroup and DeMorgan triple semigroup and by defining three Mann's compositions and one unary operation on the set of 3-place(ternary) functions over some set, we construct a DeMorgan triple semigroup of 3-place (ternary) functions and so find an abstract characterization of this algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Hv-semilattice and obtain some characterizations of it. We give the definitions of ideal and of hyperorder on an Hv-semilattice. We also study some of their related properties. [ABSTRACT FROM AUTHOR]

Abstract: The soft set theory, proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. By combining the interval-valued fuzzy set and soft set models, the purpose of this paper is to introduce the concept of the interval-valued fuzzy soft set. The complement, “AND” and “OR” operations are defined on the interval-valued fuzzy soft sets. The DeMorgan’s, associative and distribution laws of the interval-valued fuzzy soft sets are then proved. Finally, a decision problem is analyzed by the interval-valued fuzzy soft set. Some numerical examples are employed to substantiate the conceptual arguments. [Copyright &y& Elsevier]

Abstract: A class-bridge rule is the rule whose antecedent and consequent belong to different conceptual classes. This kind of rules stands for the correlation between conceptual classes. The study on class-bridge rules can benefit many domains such as criminal detection, chemical synthesis, biological grafting, etc. Class-bridge rules are different from other association rules as follows: (1) class-bridge rules can be generated from infrequent itemsets; (2) measurements of class-bridge rules include the distance between conceptual classes and the relation between the antecedents/consequents and their affiliated conceptual classes. This paper proposes an algorithm based on rough sets to mine and evaluate class-bridge rules. The experiment demonstrates its effectiveness. [Copyright &y& Elsevier]

Scholars often consider the arithmetic mean as the only mean available. This gives rise to several mistakes. Thus, in a first course in statistics, it is necessary to introduce them to a more general concept of mean. In this work we present the notion of mean suggested by Oscar Chisini in 1929, which has a double advantage. It focuses students' minds on the substance of the problem for which a mean is required, thus discouraging any automatic procedure, and it does not require a preliminary list of the different mean formulas. Advantages and limits of the Chisini mean are discussed by means of examples. [ABSTRACT FROM AUTHOR]

We consider algebras with a single n-ary operation and a certain type of associativity. We prove that (up to isomorphism) there are exactly 5 of these associative mono-n-ary algebras with 2 elements for even n ≥ 2 and 6 for odd n ≥ 3. These algebras are described explicitly. It is shown that a similar result is impossible for algebras with at least 4 elements. An application concerning the assignment of a control bit to a string is given. [ABSTRACT FROM AUTHOR]

A Moufang loop is a binary system that satisfies a particular weak form of the associative law. Doro and Glauberman observed that there is a direct connection between simple Moufang loops and simple groups with triality. Using this correspondence, Liebeck proved that nonassociative finite simple Moufang loops arise from split octonion algebras over finite fields. We extend Liebeck's theorem to the case of locally finite simple Moufang loops. [ABSTRACT FROM AUTHOR]

In this paper we prove that the heart of a nondegenerate Jordan system (algebra, triple system or pair) is either simple or zero. We also obtain Herstein type results relating the hearts of associative systems and those of their corresponding Jordan systems. [Copyright &y& Elsevier]

Focuses on the General Associative Law (GAL) in mathematics. Generalizations about the GAL; Different treatments of the GAL in the literature.

Presents a study which designed cellular neural networks with a stable equilibrium point to behave as associative memories. Results; Conclusion.

Presents a study which discussed an inner-product processor for the multi-valued exponential bidirectional associative memory (MV-eBAM). Impact of bidirectional associative memory to computer networks; Theory of MV-eBAM; Conclusion.

The article presents a discussion aimed at proving that Axiom 1 of the first axiom system in Alfred Tarski's book, "Introduction to Logic and to the Methodology of Deductive Sciences," can be derived from all the succeeding axioms he discussed. It is mentioned that Tarski has provided two axiom systems for the arithmetic of real numbers. It is pointed out that except for the first one, there is a mutual independence in all axioms of the first system. The properties of Tarski's axiom system is also provided. Several theorems derived from Axioms 2-9 similar to Tarski's book, are also discussed including the Commutative Law, Associative Law, Existence of a Zero Element, and the Law of Monotonicity.

This paper provides a framework that highlights the features of computer models that make them especially vulnerable to floating-point errors, and suggests ways in which the impact of such errors can be mitigated. We focus on small floating-point errors because these are most likely to occur, whilst still potentially having a major influence on the outcome of the model. The significance of small floating-point errors in computer models can often be reduced by applying a range of different techniques to different parts of the code. Which technique is most appropriate depends on the specifics of the particular numerical situation under investigation. We illustrate the framework by applying it to six example agent-based models in the literature. [ABSTRACT FROM AUTHOR]