Abstract: In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We study in detail the operation in $\mathcal{L}^p$ spaces and in sets of continuous functions, in a different way to previous works of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.

Abstract: We study the lattice of varieties of monoids, i.e., algebras with two operations, namely, an associative binary operation and a 0-ary operation that fixes the neutral element. It was unknown so far, whether this lattice satisfies some non-trivial identity. The objective of this paper is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids, and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.

Abstract: A structural theory of operations between real-valued (or extended-real-valued) functions on a nonempty subset A of Rn is initiated. It is shown, for example, that any operation ∗ on a cone of functions containing the constant functions, which is pointwise, positively homogeneous, monotonic, and associative, must be one of 40 explicitly given types. In particular, this is the case for operations between pairs of arbitrary, or continuous, or differentiable functions. The term pointwise means that (f∗g)(x) = F (f(x), g(x)), for all x ∈ A and some function F of two variables. Several results in the same spirit are obtained for operations between convex functions or between support functions. For example, it is shown that ordinary addition is the unique pointwise operation between convex functions satisfying the identity property, i.e., f ∗ 0 = 0 ∗ f = f , for all convex f , while other results classify Lp addition. The operations introduced by Volle via monotone norms, of use in convex analysis, are shown to be, with trivial exceptions, precisely the pointwise and positively homogeneous operations between nonnegative convex functions. Several new families of operations are discovered. Some results are also obtained for operations that are not necessarily pointwise. Orlicz addition of functions is introduced and a characterization of the Asplund sum is given. A full set of examples is provided showing that none of the assumptions made can be omitted.

Abstract: In this paper, in the first we introduce the concept of algebra of the power set and new notions connected to it are investigated and discussed like subalgebra of the power set, soft algebra of the power set and soft subalgebra of the power set. Then some binary operations between two soft algebras of the power set are studied. Further, we state the relations between soft algebra of the power set and soft algebra of the power set. Moreover, several examples are given to illustrate the notations introduced in this work.

Abstract: Memory and computation efficient deep learning architec- tures are crucial to continued proliferation of machine learning capabili- ties to new platforms and systems. Binarization of operations in convo- lutional neural networks has shown promising results in reducing model size and computing efficiency. In this paper, we tackle the problem us- ing a strategy different from the existing literature by proposing local binary pattern networks or LBPNet, that is able to learn and perform binary operations in an end-to-end fashion. LBPNet1 uses local binary comparisons and random projection in place of conventional convolu- tion (or approximation of convolution) operations. These operations can be implemented efficiently on different platforms including direct hard- ware implementation. We applied LBPNet and its variants on standard benchmarks. The results are promising across benchmarks while provid- ing an important means to improve memory and speed efficiency that is particularly suited for small footprint devices and hardware accelerators.

Abstract: Let be the algebra of all bounded linear operators on Banach space X. We determine the form of maps (not necessarily linear) which satisfies the following condition for different kinds of binary operations on operators: The sum , the product AB, triple product ABA, and Jordan product for all where is the set of operators commuting with .

Abstract: By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.

Abstract: University mathematics students often find the content of linear algebra difficult because of the abstract and highly theoretical nature of the subject as well as the formal logic required to carry out proofs. This chapter explores some specific difficulties experienced by students when negotiating vector space and subspace concepts. Seventy-three in-service mathematics teachers’ responses to two items testing the ability to prove that a given set is not a subspace and that a given set is a subspace of a vector space were studied in detail. Follow-up interviews on the written work were conducted to identify the participants’ ways of understanding. The action–process–object–schema (APOS) theory was used to unpack the structure of the concepts. Findings reveal that the teachers struggled with the vector sub-space concepts mainly because of prior non-encapsulation of prerequisite concepts of sets and binary operations and difficulties with understanding the role of counter-examples in showing that a set is not a vector subspace.

Abstract: The superextension $\lambda(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: X\times X \to X$ can be extended to an associative binary operation $*: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $\leq 5$.

Abstract: A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-free-path finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jiraskova and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic union-free languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each \(n \geqslant 3\) there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has \(n^n\) elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.

Abstract: We discuss the generalization, in higher dimensions, of the tropical semiring, whose two binary operations on the set of real numbers together with infinity are defined to be the minimum and the sum of a pair, respectively. In particular, our objects are closed convex sets, and for any pair, we take the convex hull of their union and their Minkowski sum, respectively, as the binary operations. We consider the semiring in several different cases, determined by a recession cone.

Abstract: Let be the space of complex matrices, and for a nonzero vector and , let denote the local spectrum of T at e. Characterizations are obtained for maps on satisfying where the binary operation stands for the skew Jordan semi-triple product or the skew product on matrices, with no additional assumption on the range of the map .

Abstract: In this paper we define a new product-like binary operation on directed graphs, and we discuss some of its properties. We also briefly discuss its application in constructing the subtyping relation in generic nominally-typed object-oriented programming languages.

Abstract: We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.

Abstract: We define and study binary operations for homotopy groups with coefficients, and give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of the generalized Whitehead product to these operations. We discuss two classes of binary operations, i.e., the Whitehead products and the torsion products. We also introduce a new class of operations called Ext operations and determine some of its properties. Then we compare the torsion product to the Whitehead product in a special case, and prove that the smash product of two Moore spaces has the homotopy type of a wedge of two Moore spaces.

Abstract: In this paper, continuous binary operations of a topological space are studied and a criterion of their invertibility is proved. The classification problem of groups of invertible continuous binary operations of locally compact and locally connected spaces is solved. A theorem on the binary distributive representation of a topological group is also proved.

Abstract: Using the technique of higher derived brackets developed by Voronov, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic $2$-manifold. The algebra we obtain has a particularly nice structure, in that it accommodates the Dorfman bracket of a Courant algebroid as the binary operation in the hierarchy of operations, and the defect in the symmetry of each operation is measurable in a certain precise sense. We move to call such an algebra a homotopy Dorfman algebra, or a $D_\infty$-algebra, which leads to the construction of a homotopy Courant algebroid.

Abstract: In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, uniform spaces, partial metric spaces, $b$-metric spaces etc. In this context, first we define a binary normed operation on nonnegative real numbers and give some examples. Then we recall the concept of $T$-metric space and some important and fundamental properties of it. A $T$-metric space is a $3$-tuple $(X, T, \diamond)$, where $X$ is a nonempty set, $\diamond$ is a binary normed operation and $T$ is a $T$-metric on $X$. Since the triangular inequality of $T$-metric depends on a binary operation, which includes the sum as a special case, a $T$-metric space is a real generalization of ordinary metric space. As main results, we present three coupled fixed point theorems for bivariate mappings satisfying some certain contractive inequalities on a complete $T$-metric space. It is easily seen that not only existence but also uniqueness of coupled fixed point guaranteed in these theorems. Also, we provide some suitable examples that illustrate our results.

Abstract: We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.

Abstract: In this paper we develop on abstract system: viz Boolean-like algebra and prove that every Boolean algebra is a Boolean-like algebra. A necessary and sufficient condition for a Boolean-like algebra to be a Boolean algebra has been obtained. As in the case of Boolean ring and Boolean algebra, it is established that under suitable binary operations the Boolean-like ring and Boolean-like algebra are equivalent abstract structures.

Abstract: The study of maps which act on a finite set of objects is of special importance in modern algebra. The main objective of this paper is to compare between the structures of groups that yield from the definition of composite functions in group theory (permutation groups) and those in calculus. In calculus there no detailed study for groups, but we took the definition of composite functions as a binary operation to define a group. This study can be considered as an essential model for isomorphic groups.

Abstract: Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible elements. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the structure of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Conditions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2,2,1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra.

Abstract: It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In our previous papers we introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study a bit more stronger version of an algebra where the binary operation is even monotonous. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

Abstract: We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.

Abstract: Abstract Let G be a simple graph. A set D ⊆ V (G) is an interior dominating set of G if D is a dominating set of G and every vertex v ∈ D is an interior vertex of G. The minimum cardinality of an interior dominating set of G, denoted by γId(G), is called an interior domination number of G. In this paper, we characterize the interior dominating sets of the join, corona, lexicographic and Cartesian products of graphs and determine the corresponding interior domination number of these graphs.