Menger algebras of \(k\)-commutative \(n\)-place functions
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 congruence on parent induces a nontrivial congruence on \({C_A}\)). The set of constants and dense embeddings are used to present a description of all Menger algebras isomorphic to some \((\mathfrak{M}_n^k(A),{\mathcal O})\) and to prove, that all automorphisms of \((\mathfrak{M}_n^k(A),{\mathcal O})\) are inner.
scientific article; zbMATH DE number 7372694
scientific article