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Representations of Menger \((2,n)\)-semigroups by multiplace functions. (English) Zbl 1092.20051

Let \({\mathcal F}(A^n,A)\) be the set of all partial mappings from \(A^n\) into \(A\). A Menger \((2,n)\)-semigroup of \(n\)-place functions is a nonempty set \(\Phi\subset{\mathcal F}(A^n,A)\) considered with respect to the following operations: an \((n+1)\)-ary operation of superposition \([\;]\) and \(n\) binary compositions \(\oplus_i\); \(i=\overline{1,n}\), defined by: \[ [f,g_1,\dots,g_n](a_1^n)=f(g_1(a_1^n),\dots,g_n(a_1^n)),\qquad (f\oplus_ig)(a_1^n)=f(a_1^{i-1},g(a_1^n),a_{i+1}^n), \] where \(a_1,\dots,a_n\in A\); \(f,g,g_1,\dots,g_n\in\Phi\) (here \(x_i^j\) denotes the sequence \(x_i,x_{i+1},\dots,x_j)\).
In the paper under review, the authors study some methods of representations of such Menger \((2,n)\)-semigroups by \(n\)-place functions and give an abstract characterization of the class of algebras \((\Phi,[\;],\oplus_1,\dots,\oplus_n,\subset)\) closed with respect to the set-theoretic inclusion.

MSC:

20N15 \(n\)-ary systems \((n\ge 3)\)
08A62 Finitary algebras
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