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Fundamentals of derivations on (ordered) hyper(near)-rings. (English) Zbl 1477.16063

The authors present the following in what they called their subhyperrings of hyperring analysis in the two order matrix ground with some sort of derivation.
1.
In any hypernear-ring with a derivation, the elements with derivative \(0\) form a subhypernear-ring.
2.
\(d\) is a derivation on a reduced hypernear-ring. Then for any subset \(B\) of it, \(d(\mathrm{Ann}(B))\subseteq\mathrm{Ann}(B)\)
3.
In \((R,+,\cdot,\leq)\), an ordered Krasner hyperring if \(d:R\to R\) is a derivation, then the map \(\Theta:M(R)\to M(R)\) defined by \(\Theta(A)=\left(\begin{smallmatrix}d(a_{11})&d(a_{12})\\ 0&0\end{smallmatrix}\right)\), where \(A=\left(\begin{smallmatrix}a_{11}&a_{12}\\ 0&0\end{smallmatrix}\right)\in M(R)\), is a derivation on \(M(R)\).
4.
In an ordered Krasner hyperring the corresponding two by two matrices with second row zeros is again an ordered Krasner hyperring with respect to corresponding operations together with the respective ordering. And then a strong \(h\)-derivation is an \(h\)-derivation.
5.
The final result is a generalization of a result to ordered additive-multiplicative hyperrings of the type \((R,+,\cdot,\leq)\) with a homomorphism \(\phi_d:R\to M'(R)\), where each element \(x\) corresponds to a matrix \((a_{ij})\) with \(a_{ii}=x\), \(a_{12} = d(x)\) and \(a_{20} =0\), where \(d\) is a self-map. Then the derivation character of a self-map on \(R\) is equivalent to the homomorphism character of \(\phi_d\) described above.
In case of an ordered additive-multiplicative hyperring, the (not a) collection of all \(2\times 2\) matrices with elements from \(R\), with the usual matrix multiplication and addition, together with respective matrix order, is an ordered additive-multiplicative hyperring. And this justifies the existence of additive-multiplicative hyperring of two order matrices admitting \(h\)-derivation with an odd degree of nilpotency.

MSC:

16Y99 Generalizations
16W25 Derivations, actions of Lie algebras
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