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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 $$x \in J$$.

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