Über dichte Ordnungstypen.
We characterize for which ultrafilters on is the ultrafilter extension of the asymptotic density on natural numbers -additive on the quotient boolean algebra or satisfies similar additive condition on . These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name (null) and (*). We also present a characterization of a - and semiselective ultrafilters...
In this paper we consider the Aleksandrov equation f(L + x) = f(L) + f(x) where L is contained in Rn and f: L --> R and we describe the class of solutions bounded from below, with zeros and assuming on the boundary of the set of zeros only values multiple of a fixed a > 0. This class is the natural generalization of that described by Aleksandrov itself in the one-dimensional case.
Our purpose is to introduce the W-composition, W-minimalization and W-primitive recursion operations as operations between W-valued functions, where W denotes the ordered semiring ([0,1],+,≤). We prove that: 1) the set of W-calculable functions is closed under the W-composition and W-primitice recursion operations, and 2) the set of the partially W-calculable functions is closed under the W-minimalization operation.