On \(N\)-summations. I (Q1610286)

A cone semiring is an additively commutative semiring \((C,+,\cdot)\) with absorbing zero 0 and identity together with a partial order \(\leq\) on \(C\) such that addition and multiplication are monotone and 0 is the smallest element in \(C\). For an infinite class \(N\) define \(C^N=\{\alpha\colon N\to C\mid\text{supp}(\alpha)=\{n\in N\mid\alpha(n)\not=0\}\) is a set\(\}\). Then \(C^N\) with pointwise operations is a left \(C\)-semimodule. Now, an \(N\)-summation for \(C\) is a pair \((S_C,\sum_C)\) where \(S_C\) is a \(C\)-subsemimodule of \(C^N\), and \(\sum_C\colon S_C\to C\) is a \(C\)-homomorphism satisfying some natural conditions. One main result is: If \(C\) is \(N\)-order complete and \(N\)-joins are compatible with addition and multiplication in \(C\), then there is an \(N\)-summation for the \(C\)-subsemimodule \(SB_C\) of all summarily bounded elements of \(C^N\). The other main result concerns an \(N\)-summation on the class of Cauchy elements of \(M^N\) for certain prenormed semitopological left semimodules \(_RM\) over a prenormed semiring \((R,+,\cdot)\) with value cone \(C\). (Also submitted to MR).