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Semigroup presentations have been studied for a long period. Since 1936 J. A. and H. S. M. have created Todd -- Coxeter algorithm for solving enumeration problems of presentations. Here we present some semigroup presentations in terms of generators and defining relations. The aim of this paper is to investigate some properties about the relationship among abstract generators defined on semigroup, using computational methods and to study relationship between some semigroup and group presentation. [ABSTRACT FROM AUTHOR]

Abstract Conceptual knowledge of key principles underlying arithmetic is an important precursor to understanding algebra and later success in mathematics. One such principle is associativity, which allows individuals to solve problems in different ways by decomposing and recombining sub-expressions (e.g. 'a + b − c' = 'b − c + a'). More than any other principle, children and adults alike have difficulty understanding it, and educators have called for this to change. We report three intervention studies that were conducted in university classrooms to investigate whether adults' use of associativity could be improved. In all three studies, it was found that those who first solved inversion problems (e.g. 'a + b − b') were more likely than controls to then use associativity on 'a + b − c' problems. We suggest that 'a + b − b' inversion problems may either direct spatial attention to the location of 'b − c' on associativity problems, or implicitly communicate the validity and efficiency of a right-to-left strategy. These findings may be helpful for those designing brief activities that aim to aid the understanding of arithmetic principles and algebra. Highlights • Associativity is an arithmetic concept that children and adults struggle to apply. • Inversion is a simpler concept that older children and adults easily apply. • Solving inversion problems ('a + b – b') increased subsequent associativity use. • 'a + b – b' inversion, but not 'a + b – a' inversion, increased associativity use. • 'a + b – b' problems may direct attention or validate a 'right-to-left' strategy. [ABSTRACT FROM AUTHOR]

Let R k , m be the ring 𝔽 2 m [ u 1 , ... , u k ] 〈 u i 2 , u i u j − u j u i 〉 . We study cyclic codes of arbitrary length n over the ring R 3 , m and obtain a complete characterization for them by presenting a unique set of generators for these codes. For example, we try to determine all distinct cyclic codes of length 2 over the ring R 3 , m by a practical algorithm. [ABSTRACT FROM AUTHOR]

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]

We extend classical Propositional Logic (PL) by adding a new primitive binary connective φ∣ψ, intended to represent the 'superposition' of sentences φ and ψ, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form M,f, where M is an ordinary two-valued assignment for the sentences of PL and f is a choice function for all pairs of classical sentences. In the new semantics φ∣ψ is strictly interpolated between φ∧ψ and φ∨ψ. By imposing several constraints on the choice functions, we obtain corresponding notions of logical consequence relations and corresponding systems of tautologies, with respect to which ∣ satisfies some natural algebraic properties such as associativity, closedness under logical equivalence and distributivity over its dual connective. Thus various systems of Propositional Superposition Logic (PLS) arise as extensions of PL. Axiomatizations for these systems of tautologies are presented and soundness is shown for all of them. Completeness is proved for the weakest of these systems. For the other systems completeness holds if and only if every consistent set of sentences is extendible to a consistent and complete one, a condition whose truth is closely related to the validity of the deduction theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to propose an operational idea for developing algebraic thinking in the absence of alphanumeric symbols. The paper reports on a design experiment encouraging preschool children to use the associative property algebraically. We describe the theoretical basis of the design, the tasks used, and examples of algebraic thinking in 5-6-year-old children. Theoretically, the paper makes a critical distinction between operational and structural meanings of the notion of equality. We argue that mathematical thinking involving equality among young learners can comprise both an operational and a structural conception and that the operational conception has a side that is productively linked to the structural conception. Using carefully designed hands-on tasks, the crux of the paper is the realization of algebraic thinking (in verbal mathematics) as operationally experienced in the ability to transform one number structure, with a quantity that is subject to change, into another through equality-preserving transformations. [ABSTRACT FROM AUTHOR]

The concept of a central soft set related with some common decision making problems in real life is introduced in this paper. Some basic operations on central soft sets and theirs properties such as distributive law and associative law are given. The study also shows that some classic operations between soft sets can be obtained by central soft sets with selecting different central sets. In the part of application, the concepts of an evaluation system for a parameter set and desired objects of a central soft set are initiated. An algorithm is presented to solve such decision making problems. [ABSTRACT FROM AUTHOR]

Construction of new associative, commutative and increasing operations on the unit interval from a given operation ⁎ is proposed. It is shown that when ⁎ is a t-norm, t-conorm or a proper uninorm, new t-norms, t-conorms and proper uninorms can be obtained. [ABSTRACT FROM AUTHOR]

The concept of a central soft set related with some common decision making problems in real life is introduced in this paper. Some basic operations on central soft sets and theirs properties such as distributive law and associative law are given. The study also shows that some classic operations between soft sets can be obtained by central soft sets with selecting different central sets. In the part of application, the concepts of an evaluation system for a parameter set and desired objects of a central soft set are initiated. An algorithm is presented to solve such decision making problems. [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]

The main motivation behind this paper is to study some structural properties of a non-associative structure (LAsemigroup) in terms of fuzzy sets as it hasn't attracted much attention compared to associative structures. An LA-semigroup can be referred to as a non-associative semigroup, as the main difference between a semigroup and an LA-semigroup is the switching of an associative law. In this paper, we introduce and characterize fuzzy (0, 2)-ideals, fuzzy (0, 2)-bi-ideals and fuzzy (1, 2)-ideals of an LA-semigroup. We give a new and alternate definition for a strongly regular element of a unitary LA-semigroup and shownovel conditions under which a strongly regular LA-semigroup becomes an LA**-semigroup. These characterizations do not exist in literature before this work. As an application of our results we get characterizations of a strongly regular LA-semigroup in terms of fuzzy one-sided ideals and fuzzy bi-ideals. Finally we introduce and investigate the properties of fuzzy inner-unitary LA-subsemigroups in an LA**-semigroup. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works. [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]

A consistent connective system generated by nilpotent operators is not necessarily isomorphic to the Łukasiewicz system. Using more than one generator function, consistent nilpotent connective systems (so-called bounded systems) can be obtained with the advantage of three naturally derived negation operators and thresholds. In this paper, equivalences in bounded systems are examined. Here, three different types of operators are studied, and a paradox of the equivalence (i.e. there is no equivalence relation in a non-Boolean setting which fulfils ∀ x e ( x , x ) = 1 and e ( x , n ( x ) ) = 0 ) is resolved by aggregating the implication-based equivalence and its dual operator. We will also show that the aggregated equivalence has nice properties like associativity, threshold transitivity and T-transitiviy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-user multichannel rendezvous problem in a cognitive radio network (CRN) and derive tight lower bounds for maximum time-to-rendezvous (MTTR) and maximum conditional time-to-rendezvous (MCTTR) of various channel hopping (CH) schemes under a channel loading constraint. In the symmetric and synchronous setting, we propose a novel Cycle-Adjustable Channel Hopping (CACH) scheme to achieve the MTTR lower bound (when the channel loading is bounded above by 1/u with u being a prime power). Thus, the MTTR lower bound is tight and the CACH scheme is optimal in minimizing MTTR among all the symmetric and synchronous CH schemes under the same channel loading constraint. In the asymmetric setting, we show that the classical wait-for-mommy strategy can be used to achieve the MCTTR lower bound, and thus it is optimal. In the symmetric and asynchronous setting, we also show a hierarchical construction of an asynchronous CH sequence by using two smaller asynchronous CH sequences. To further understand the effect of channel loading to the other performance metrics in a CRN, we perform various computer simulations for various CH schemes. Our simulation results show that the average time-to-rendezvous of CACH is independent of the total number of channels, and it is also robust to the disturbance of primary users. [ABSTRACT FROM PUBLISHER]

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 the present paper we study the associativity law of Tarski and the cyclic associativity law. We characterize the power series solutions of these equations in the complex domain and we give the connection to the solutions of the ordinary associativity equation. [ABSTRACT FROM AUTHOR]

A formal power series $${F(x_{1}, \ldots, x_{n})\in\mathbb{C}[\![x_1,\ldots x_n]\!]}$$ of order at least 1 is called n-associative, n ≥ 3, if This notion generalizes associativity which is the special case of n = 2. We determine the set of all n-associative formal power series over $${\mathbb{C}}$$ , all convergent n-associative power series, and all commutative (or symmetric) n-associative formal power series. Moreover we study relations between n- and m-associativity for certain $${n,m\in\mathbb{N}}$$ and determine the structure of associative families ( F) of formal power series $${F(x_{1}, \ldots, x_{n})\in\mathbb{C}[\![x_1,\ldots x_n]\!]}$$ . [ABSTRACT FROM AUTHOR]

Abstract. This article describes the lattice of varieties generated by those of flocks and near heaps. Flocks and heaps are two ways of presenting groups by a ternary operation rather than a binary one. Their varieties joined with that of ternary semilattices create the varieties of near flocks and near heaps. This is done by finding normal forms for words that make up free algebras. Simple sets of identities define these varieties. Identities in general are decidable. Each near flock is a Płonka sum of flocks, and each near heap is a Płonka sum of heaps. An algorithm translates any binary group identity to one in a ternary operation satisfied by near heaps. [ABSTRACT FROM AUTHOR]

Following an approach developed by Niemenmäa and Kepka, we prove that if a division groupoid G satisfies the identity Rx1Lx2...Rx2n-1Lx2ny = Lx2nRx2n-1...Lx2Rx1y for some n ≠ 2, then G is an abelian group. Using equational reasoning, we also give a new proof of a result of Niemenmäa and Kepka that a division groupoid in which the generalized associative law x1(x2(... xn-1xn)...)) = ((...(x1x2)...)xn-1)xn holds must be a group. [ABSTRACT FROM AUTHOR]

There has been a steady rise in the support for games as learning environments. This support is largely based on the strong levels of engagement and motivation observed during gameplay. What has proven difficult is the ability to consistently design and develop learning games that are both engaging and educationally viable. Those in the game-based learning community contend the solution is to tightly integrate instructional objective(s) within the game's mechanics. In order to investigate whether or not intrinsic integration would aid in game design, an experimental study was conducted with 111 fourth and fifth graders in which three versions of a game were designed in order to determine which approach would have a more significant impact on conceptual understanding of the associative and distributive properties. It was found that those who played the intrinsic version of the game (properties mapped to game mechanics) had significant learning gains compared with those who played the extrinsic version. Perceptions of the game are also discussed, along with future directions. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced the notion of neutrosophic (2, 2) -regular, neutrosophic strongly regular neutrosophic AG -groupoids and investigated these structures. We have shown that neutrosophic regular, neutrosophic intra-regular and neutrosophic strongly regular AG -groupoid are the only generalized classes of neutrosophic AG -groupoid. Further we have shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, fx completely regular, (2, 2) -regular and strongly regular AG* - neutrosophic groupoids do not exist. [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]

Computations in orthomodular lattices are rather difficult because there is no known analogue of the standard disjunctive or conjunctive forms from boolean algebras. Tools admitting simplifications of formulas are rather limited. In previous work, we investigated the possibility of finding associative operations in orthomodular lattices that would admit new techniques of theorem proving. It was found that there are only 6 operations in orthomodular lattices which are (already known to be) associative. In this paper, we replaced associativity by more general properties, in particular by the identities that hold in alternative algebras. We have found that there are 8 nonassociative operations in orthomodular lattices satisfying the identities of alternative algebras. For these operations, we clarified the validity of other similar identities, including the Moufang identities. We hope that this effort can contribute to clarifying to what extent the validity of identities in orthomodular lattices can be algorithmized. [ABSTRACT FROM AUTHOR]

We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Diffie-Hellman algorithm.We prove the existence of non-associative groupoidswhich are simultaneously power commuting and not power-associative. [ABSTRACT FROM AUTHOR]

The article offers information about the associative property, one of the methods developed linking single-period arithmetic attribution results. It reveals that associative property renders the Menchero and the Ad Hoc methods invalid and discusses that non-associative methods violate another vital property which refers to a sign preservation. Other information such as the use of the Frongello methods is presented.

We study pseudocompact primitive topological inverse semigroups, describe the structure of pseudocompact primitive topological inverse semigroups, and show that the Tychonoff product of a family of pseudocompact primitive topological inverse semigroups is a pseudocompact topological space. Moreover, we prove that the Stone-Čech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup. [ABSTRACT FROM AUTHOR]

We discuss associative analogues of classical Yang-Baxter equation (CYBE) meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang-Baxter equations (AYBE). [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]

Highlights: [•] We introduce a novel non-linear forecasting technique based on the Gamma classifier. [•] Its performance for long-term time horizons was tested on synthetic and real data. [•] Two benchmark time series were used for testing. [•] Six time series related to monthly oil production were also used. [•] The Gamma classifier model outperformed previous techniques in forecast accuracy. [Copyright &y& Elsevier]

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]

For the stochastic multi-criteria decision-making problem, in which the information on criteria's weights is incomplete and the indices value of alternatives are in the form of intuitionistic fuzzy numbers, an intuitionistic random decision-making approach based on MYCIN certainty factor and prospect theory is proposed. According to the score function of intuitionistic fuzzy numbers and prospect theory, the MYCIN certainty factors of different alternatives in different indices are obtained. And the trust degrees of different indices are determined by using the grey incidence analysis. Then, the certainty factor fusion method in different indices is deduced to get the optimal alternative. It is proved that the fusion method satisfies the commutative law and the associative law. And the best alternative is got by using the method. Finally, an example shows the feasibility and validity of this method. [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]

In this paper the notion of bipolar-valued fuzzy LA-subsemigroups is introduced and also some properties of bipolar- valued fuzzy left (right, bi-, interior) ideals of LA-semigroups has been discussed. [ABSTRACT FROM AUTHOR]