scholarly journals Ternary Menger Algebras: A Generalization of Ternary Semigroups

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 553
Author(s):  
Anak Nongmanee ◽  
Sorasak Leeratanavalee
Keyword(s):  
Natural Number ◽  
Homomorphism Theorem ◽  
Menger Algebra ◽  
Suitable Generalization ◽  
Ternary Semigroup

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.

scholarly journals Band description of knots and Vassiliev invariants

2002 ◽  
Vol 133 (2) ◽  
pp. 325-343 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
AKIRA YASUHARA
Keyword(s):  
Natural Number ◽  
Algebraic Condition ◽  
Vassiliev Invariants

In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k−1.

scholarly journals Another characterization of congruence distributive varieties

2020 ◽  
Vol 57 (3) ◽  
pp. 284-289
Author(s):  
Paolo Lipparini
Keyword(s):  
Natural Number ◽  
Congruence Identity ◽  
Congruence Distributive

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

scholarly journals Finding Unavoidable Colorful Patterns in Multicolored Graphs

10.37236/8184 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Matt Bowen ◽  
Ander Lamaison ◽  
Alp Müyesser
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Type Problem ◽  
Polish Spaces ◽  
Subgraph Isomorphic ◽  
Simple Characterization ◽  
Color Class ◽  
Ramsey Type

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre. 

scholarly journals On Characterization of Open Sets in Euclidean Spaces

2018 ◽  
Vol 22 ◽  
pp. 01007
Author(s):  
Firudin Kh. Muradov
Keyword(s):  
Topological Spaces ◽  
Euclidean Spaces ◽  
Abstract Characterization ◽  
Ternary Semigroup ◽  
Associative Law ◽  
Open Maps

A ternary semigroup is a nonempty set T together with a ternary oper- ation [abc] satisfying the associative law [[abc] de] = [a [bcd] e] = [ab [cde]] for all a, b, c, d, e ε T. A map f between topological spaces X and Y is called open if the image of each set open in X is open in Y. The pur- pose of this paper is to give an abstract characterization of the ternary semigroups of open maps defined on open sets in Euclidean n-spaces.

A characterization of lambda-terms transforming numerals

2016 ◽  
Vol 26 ◽  
Author(s):  
PAWEŁ PARYS
Keyword(s):  
Natural Number ◽  
Linear Transformation ◽  
Fixed Number ◽  
The Other ◽  
Natural Numbers ◽  
Data Types ◽  
Fixed Type ◽  
Intersection Types ◽  
The One

AbstractIt is well known that simply typed λ-terms can be used to represent numbers, as well as some other data types. We show that λ-terms of each fixed (but possibly very complicated) type can be described by a finite piece of information (a set of appropriately defined intersection types) and by a vector of natural numbers. On the one hand, the description is compositional: having only the finite piece of information for two closed λ-terms M and N, we can determine its counterpart for MN, and a linear transformation that applied to the vectors of numbers for M and N gives us the vector for MN. On the other hand, when a λ-term represents a natural number, then this number is approximated by a number in the vector corresponding to this λ-term. As a consequence, we prove that in a λ-term of a fixed type, we can store only a fixed number of natural numbers, in such a way that they can be extracted using λ-terms. More precisely, while representing k numbers in a closed λ-term of some type, we only require that there are k closed λ-terms M1,. . .,Mk such that Mi takes as argument the λ-term representing the k-tuple, and returns the i-th number in the tuple (we do not require that, using λ-calculus, one can construct the representation of the k-tuple out of the k numbers in the tuple). Moreover, the same result holds when we allow that the numbers can be extracted approximately, up to some error (even when we only want to know whether a set is bounded or not). All the results remain true when we allow the Y combinator (recursion) in our λ-terms, as well as uninterpreted constants.

scholarly journals Permutations that preserve asymptotically null sets and statistical convergence

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4821-4827
Author(s):  
Jeff Connor
Keyword(s):  
Natural Number ◽  
Statistical Convergence ◽  
Asymptotic Density ◽  
Arbitrary Natural Number ◽  
Convergent Sequences

The main result of this article is a characterization of the permutations ?: N ? N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p ? ? where p is an arbitrary natural number and Cp = {l : ?-1(l)? lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

scholarly journals Ternary semigroups of topological transformations

2021 ◽  
Vol 102 (2) ◽  
pp. 84-91
Author(s):  
F.Kh. Muradov ◽  
Keyword(s):  
Euclidean Spaces ◽  
Finite Dimensional ◽  
Ternary Operation ◽  
Ternary Semigroup ◽  
Topological Transformations

A ternary semigroup is a nonempty set with a ternary operation which is associative. The purpose of the present paper is to give a characterization of open sets of finite-dimensional Euclidean spaces by ternary semigroups of pairs of homeomorphic transformations and extend to ternary semigroups certain results of L.M. Gluskin concerned with semigroups of homeomorphic transformations of finite-dimensional Euclidean spaces.

Menger systems of idempotent cyclic and weak near-unanimity multiplace functions

2021 ◽  
Author(s):  
Thodsaporn Kumduang ◽  
Sorasak Leeratanavalee
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Strong Connection ◽  
Abstract Characterization ◽  
Menger Algebra ◽  
Specific Formula ◽  
Necessary And Sufficient ◽  
Near Unanimity

Multiplace functions, which are also called functions of many variables, and their algebras called Menger algebras have been studied in various fields of mathematics. Based on the theory of many-sorted algebras, the primary aim of this paper is to present the ideas of Menger systems and Menger systems of full multiplace functions which are natural generalizations of Menger algebras and Menger algebras of [Formula: see text]-ary operations, respectively. Two specific types of [Formula: see text]-ary operations, which are called idempotent cyclic and weak near-unanimity generated by cyclic and weak near-unanimity terms, are provided. The Menger algebras under consideration have a two-element universe, the elements of which are two specific [Formula: see text]-ary operations. Additionally, we provide necessary and sufficient conditions in which the abstract Menger algebra and the Menger algebras of these two [Formula: see text]-ary operations are isomorphic. An abstract characterization of unitary Menger systems via systems of idempotent cyclic and weak near-unanimity multiplace functions is generally investigated. A strong connection between clone of terms and Menger systems of full multiplace functions is also investigated.

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