Multiplace Functions Research Articles

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

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.

Menger hyperalgebras and their representations

The paper is devoted to the investigation of algebraic hyperstructures. The concept of Menger hyperalgebras, which is a canonical generalization of semihypergroups, is introduced. The emphasis of this paper is on the algebraic nature of such structure concerning subhyperalgebras, homomorphisms and quotient hyperstructures, that allows for a rich algebraic theory. Based on the theory of multiplace functions, multivalued full functions (or hyperoperation) on a general fixed arity is defined. This leads us to construct the Menger hyperalgebras of multivalued full n-ary functions. In particular, we prove that every abstract Menger hyperalgebra can be represented by multivalued full n-ary functions.

Closed classed of three-valued logic that contain essentially multiplace functions

AbstractAll 23 closed classes of three-valued logic that contain essentially multiplace linear functions are defined. It is shown that the order of every such class is not greater than 3.

Closed classed of three-valued logic that contain essentially multiplace functions

Определяются все 23 замкнутых класса трехзначной логики, которые содержат существенно многоместные линейные функции. Устанавливается, что порядок каждого их этих классов не превосходит 3. Работа выполнена при поддержке Российского фонда фундаментальных исследований, проект 13-01-00958.

Stationary Subsets of Functional Menger ∩-Algebras of Multiplace Functions

A functional Menger ∩-algebra is a set of n-place functions containing n projections and closed under the so-called Menger's composition of n-place functions and the set-theoretic intersection of functions. We give the abstract characterization for these subsets of functional Menger ∩-algebras which contain functions with fixed points.

Representations of (2, n )-semigroups by multiplace functions

We describe the representations of (2, n )-semigroups, i.e., sets with n binary associative operations, by partial n -place functions and prove that any such representation is a union of some family of representations induced by Schein’s determining pairs.

Representations of Menger (2, n)-Semigroups by Multiplace Functions

ABSTRACT Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics, where we consider various operations on sets of functions, which are naturally defined. The basic operation for n -place functions is an (n + 1)-ary superposition [ ], but there are some other naturally defined operations, which are also worth of consideration. In this article we consider binary Mann's compositions ⊕1,…,⊕ n for partial n-place functions, which have many important applications for the study of binary and n-ary operations. We present methods of representations of such algebras by n-place functions and find an abstract characterization of the set of n-place functions closed with respect to the set-theoretic inclusion. Communicated by V. Artamonov.

FUNCTIONAL MENGER -ALGEBRAS

ABSTRACT Investigation of multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for functions is superposition (composition), but there are some other naturally defined operations, which are also worth of consideration. For example, the operation of set-theoretic intersection and the operation of projections. In this paper we find an abstract characterization of the set of multiplace functions which are closely related to these three operations.

Stationary subsets and stabilizers of restrictive Menger P-algebras of multiplace functions

It is known [5], [6] that restrictive Menger algebras of multiplace functions are the sets of functions of fixed arity, which are stable for the operations of superposition, set-theoretical intersection and restrictive multiplication (i.e. the operation of the restriction of functions on domains of another functions). In the class of all such algebras we find the abstract characterization for sets of functions having some fixed point or the same fixed point for all functions of the given sets.

Theory of restrictive Menger algebras

In modern algebra, much attention is being paid at present to the study of superpositions of multiplace functions, which is explained by their applications to mathematics, cybernatics, programming, and other domains of science [I, 2]. In this connection, it is necessary to axiomatize certain concrete classes of algebras of multiplace functions. Side by side with the operation of superposition in function algebras, we also quite often consider other naturally defined operations, which enable us to express various dependences between the functions in convenient manner [3]. One of these operations is the operation of restrictive multiplication of functions, i.e., the operation of restricting one function to the domain of another function, considered for the first time by Vagner for one-place functions [4]. A joint study of this operation with the operation of superposition, and also with the operation of set-theoretic intersection in the case of multiplace functions, has been carried out in [5, 6], in which the restrictive Menger algebras and the restrictive Palgebras of multiplace functions have been characterized by finite systems of elementary axioms. In the consideration of sets of reversible multiplace functions, i.e., of functions that are one-to-one with respect to each variable separately, earlier only restrictive Palgebras could be characterized [6]. No characterization of the restrictive Menger algebras of reversible n-place functions has yet been found for n ~ 2; it is obtained in the present article.

A characterization of infimum functions

An abstract characterization of multi-place infimum functions on meet-semilattices is given.

Axiomatization of polynomial substitution algebras

The axiomatization of algebras of functions or partial functions under various operations has been studied by several authors. Menger [2] deals with the problem of axiomatizing the algebraic properties of 1-ary functions from the reals to the reals. He considers algebras with one of three binary operations corresponding to addition, multiplication or composition. This study is furthered by Schweizer and Sklar in [3]–[6]. In addition to the operations mentioned above they also introduce a partial ordering which corresponds to restriction of functions. In the second of these papers they give a set of axioms such that any system satisfying these axioms is order isomorphic to a concrete system of partial functions under composition and ordered by restriction. In [7] Schweizer and Sklar extend their work to the algebra of multiplace vector-valued functions. In [8] Whitlock studies abstract multiplaced function systems given by super-associative laws and shows that these systems are isomorphically embeddable in a concrete system of multiplaced functions where the operation on the functions is substitution of an m-ary function in an n-ary function.In this paper we consider the set of functions from Uα, the set of α-sequences, to U where a is an infinite ordinal. As opposed to the composition operations studied in the above works, we consider the composition operations *κ, for κ < α, in a narrow sense—the substitution of one function in the κth place of another. The algebras studied have the form ‹A, *κ, Vκ› κ<α where Vκ are the projection (selector) functions on the κth place. In particular, the polynomials over an algebra form such an algebra called a polynomial substitution algebra. In §6 we show that a first-order axiom system and a condition of local finiteness, given by Pinter in a talk at Berkeley in 1972 to characterize term substitution abgebras, also characterize these polynomial substitution algebras.

Algebras of multiplace functions

Algebras of multiplace functions

Connections between projective geometry and superassociative algebra

This paper establishes a connection between projective geometry and the superassociative algebra of multiplace functions. In Part I a quaternary operation is defined for the points on a line j in a projective plane π relative to a fixed quadrangle and the result of operating on A,B,C,D is denoted by A(B,C,D). It is shown that π is a Pappus plane if and only if this operation is superassociative: (A(B,C,D))(E,F,G)=A(B(E,F,G),C(E,F,G),D(E,F,G)) for any points A,B,C,D,E,F,G on j. Desargues and certain special Desargues planes are also characterized by restrictions of the superassociative law. In Part II projective geometry is developed over superassociative systems.

Indeterminate forms for multi-place functions

Indeterminate forms for multi-place functions

A composition algebra for multiplace functions

A composition algebra for multiplace functions