Multistructures determined by differential rings. (English) Zbl 1090.12500

The aim of this paper is to construct some multistructures on differential rings in a natural way, to show some properties of this multistructures, and at the conclusion to apply these results to rings of continuously differentiable functions.
Briefly, the multistructures (hypergroupoids) are pairs \((M,.)\), where \(M\) is a nonempty set and \(.\) is an operation whose result is a nonempty subset of \(M\). If a hypergroupoid satisfies the reproduction axiom \((a. M = M = M.a\;\text{for each}\;a \in M)\), then it is called a quasi-hypergroup. A quasi-hypergroup is said to be a hypergroup if the multioperation is associative.
It is shown in this paper that for a differential ring \((R,+,.,\Delta _R)\) the multistructure \((R,*)\) is a commutative hypergroup with further special properties on differential endomorphisms of \((R,+,.,\Delta _R)\). Here the operation \(*\) is defined for \(x,y\) by \[ x*y =\{ d_1\dots d_n (z): z \in \{x,y\}, d_k \in \Delta _k\}. \] Let \(J = (a,b)\) be an open interval of \(R\). Then for \(\varphi ,\psi \in C^{\infty }(J)\) a hyperoperation \( *_{(\varphi ,\psi )}\) on the ring \(C^{\infty }(J)\) is defined by \[ f *_{(\varphi ,\psi )} g = \int (\varphi ' (x) f(x) + \psi ' (x) g(x))\,\text dx, \qquad f,g \in C^{\infty }(J). \] In Theorem 3 it is proved that the hypergroupoid \((C^{\infty }(J), *_{(\varphi ,\psi )})\) is a quasi-hypergroup provided that \(\varphi '(x).\psi '(x) \neq 0\) for all \(x \in J\).


12H05 Differential algebra
20N20 Hypergroups
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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