On some Menger algebras of multiplace transformations of ordered sets
Let \(f[f_1 \dots f_n]\) denote the \((n+1)\)-ary Menger composition of mappings \(f,f_1 ,\dots, f_n : A^n \to A\), defined by \(f[f_1 \dots f_n] (a_1, \dots, a_n) = f(f_1(a_1, \dots, a_n), \dots, f_n(a_1, \dots, a_n))\). A mapping \(f : A^n \to A\) on an ordered set \((A; \leq)\) is called: extensive if \(a_i \leq f(a_1, \dots, a_n)\) for all \(a_1, \dots, a_n \in A\); idempotent if \(f[f \dots f] = f\); isotonic if for all \(a_1,\dots, a_n \in A\), \(u \in A^n\), \(i = 1,\dots, n\) the condition \(a_1 \leq a_2\) implies \(f(u|_i a_1) \leq f(u|_i a_2)\) (here \(f(u|_i x)\) denotes \(f(u_1, \dots, u_{i-1}, x, u_{i+1}, \dots, u_n))\); a semiclosure operation, if it is extensive and isotonic; closure operation, if it is idempotent and semiclosure. An abstract characterization is given of Menger algebras of extensive \(n\)-place mappings and \(n\)-place semiclosure operations and some results are presented concerning closure operations (the set of closure operations is not closed under Menger composition).
scientific article; zbMATH DE number 759388
scientific article