Menger algebras of \(k\)-commutative \(n\)-place functions
type: Item
instance of: scholarly article
title: Menger algebras of \(k\)-commutative \(n\)-place functions
zbMATH Open document ID: 1491.20168
author: Valentin S. Trokhimenko
author: Wiesław A. Dudek
published in: Georgian Mathematical Journal
publication date: 2021-07-19T00:00:00Z
review text: Let \({g_1},\dots ,{g_n} = g_1^n\), \(g,\dots ,g = \mathop g\limits^n \). If \({\mathcal T}({A^n},A)\) is the set of all \(n\)-place everywhere defined functions on a nonempty set \(A\) and the \(n + 1\)-ary Menger superposition operation is defined by \({\mathcal O}(f,g_{_1}^n)(x_1^n) = f(g_1^n(x_1^n))\) then every subset \(\Phi \subseteq {\mathcal T}({A^n},A)\) closed for \({\mathcal O}\) is a Menger algebra \((\Phi ,{\mathcal O})\) of rank \(n\); alternative notation for Menger superposition is \({\mathcal O}(f,g_1^n) = f[g_1^n]\). An \(n\)-place function \(f \in {\mathcal T}({A^n},A)\) is \(k\)-commutative, \(1 \leqslant k \leqslant n - 1\), if for all \(a,b \in A\) and all \(i = 1,2,\dots ,n - k\) holds\(f(\mathop a\limits^{i - 1} ,\mathop b\limits^k ,\mathop a\limits^{n - i - k + 1} ) = f(\mathop b\limits^{i - 1} ,\mathop a\limits^k ,\mathop b\limits^{n - i - k + 1} )\); \(\mathfrak{M}_n^k(A)\) is the set of all \(k\)-commutative \(n\)-place functions on the set \(A\), \((\mathfrak{M}_n^k(A),{\mathcal O})\) is a subalgebra of \(({\mathcal T}({A^n},A),{\mathcal O})\). It is shown that a Menger algebra \((G,o)\) of rank \(n \geqslant 2\) can be isomorphically embedded into a Menger algebra \((\mathfrak{M}_n^k(A),{\mathcal O})\) iff \(x[\mathop y\limits^{i - 1} \mathop z\limits^k \mathop y\limits^{n - i - k + 1} ] = x[\mathop z\limits^{i - 1} \mathop y\limits^k \mathop z\limits^{n - i - k + 1} ]\). A subset \(H\) of a Menger algebra \((G,o)\) is \begin{itemize} \item[(1)] an \(s\)-ideal iff \((\forall h,x_1^n \in G)(h \in H \Rightarrow h[x_1^n] \in H)\); \item[(2)] a \(v\)-ideal iff \((\forall x,h_1^n \in G)(h_1^n \in H \Rightarrow x[h_1^n] \in H)\); \item[(3)] an \(sv\)-ideal iff it is both an \(s\)- and a \(v\)-ideal. \end{itemize} Denote by \({C_A} = \{ {f_a}:a \in A,(\forall {a_1},\dots ,{a_n} \in A)f(a_1^n) = a\} \) the set of all constant functions. It is shown that \({C_A}\) is densely embedded in \((\mathfrak{M}_n^k(A),{\mathcal O})\) (every nontrivial co…
Mathematics Subject Classification ID: 20N15
Mathematics Subject Classification ID: 08A05
zbMATH DE Number: 7372694
zbMATH Keywords: Menger algebra
zbMATH Keywords: algebra of multiplace functions
zbMATH Keywords: \(sv\)-ideals
zbMATH Keywords: inner automorphism
reviewed by: Jaak Henno
MaRDI profile type: Publication
full work available at URL: gmj-2019-2072
OpenAlex ID: W2990603216
cites work: Regular elements of the complete semigroups of binary relations of the class Σ6(X,8)
cites work: Algebras of multiplace functions
cites work: Q2966387
cites work: On σ-commutativity in Menger algebras of n-place functions
cites work: Q5641309
cites work: Q3281106
cites work: Q4748365
DOI: 10.1515/GMJ-2019-2072
Recommended article: Menger Algebras of n-Place Functions
Recommended article: On σ-commutativity in Menger algebras of n-place functions
Recommended article: Q2966387
Recommended article: Q3408683
Recommended article: Menger algebras of \(n\)-place interior operations.
Recommended article: FUNCTIONAL MENGER -ALGEBRAS
Recommended article: On some subtraction Menger algebras of multiplace functions
Recommended article: On Menger algebras
Recommended article: Q4882574
Recommended article: Q3558112
description: scientific article; zbMATH DE number 7372694