In the operator version of the Hahn-Banach-Kantorovich theorem, the range space Y is assumed to be Dedekind complete. Y. A. Abramovich and A. W. Wickstead improved this by assuming only the Cantor property on Y . Along the same line of reasoning, we obtained in this paper several new results of this type. We also see that assuming Cantor property on the domain spaces instead gives good results,...

The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. Mor...

Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the have sets of Dirichlet L -series then they We extend this result by showing that isomorphisms between in bijection preserving character groups.

We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind–Mertens and prove the converse under suitable dimensionality restrictions.

Current theories of real numbers, measure, probability, and stochastic processes appear to be selfinconsistent. We exemplify the inconsistencies with Poisson processes, Markov chains, continuous random variables, the Lebesgue measure, and the Cantor-Dedekind theory of real numbers.

In this paper it is shown that the ordered structure of the Dedekind real numbers is effectively homogeneous in any topos with natural numbers object. This result is obtained within the framework of local set theory.

We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures. MSC: 06F05; 06F20

We show that there are exactly 2Dk inequivalent orientations of the rank-k free spike (k ≥ 4) where Dk is the k th Dedekind number. Our proof of this result is constructive and employs the monotone boolean functions.

The symmetric group S(n) is partially ordered by Bruhat order. This order is extended by L. Renner to the set of partial injective functions of {1, 2, . . . , n} (see, Linear Algebraic Monoids, Springer, 2005). This poset is investigated by M. Fortin in his paper The MacNeille Completion of the Poset of Partial Injective Functions [Electron. J. Combin., 15, R62, 2008]. In this paper we show tha...

In this paper we partially answer two open questions concerning clean rings. First, we demonstrate that if a quasi-continuous module is strongly clean then it is Dedekind-finite. Second, we prove a partial converse. We also prove that all clean decompositions on submodules of continuous modules extend to the entire module.