Search

We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the P-adic topology on R ∖ P . Under certain hypothesis, we show that we can associate to a prime ideal P of R a partially ordered set, constructed from some subgroups of the group of units of R / P n , which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of Z are the identity and the multiplication by −1. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of Q and different from Z is not homeomorphic to the Golomb space of Z .

Information systems play an important role in characterizing order structures. In this paper, we introduce the notions of the algebraic information system and algebraic L-information system. They are of the same logic-oriented style as the information system introduced by Scott (1982). But the axioms in this paper are briefer than reported in existing work. We also prove that the two new information systems exactly represent the algebraic domains and algebraic L-domains respectively. Based on the notion of approximable mapping between the algebraic information systems and the algebraic L-information systems, we obtain the result that the corresponding categories of algebraic information systems and algebraic L-information systems are equivalent to the category of algebraic domains and algebraic L-domains respectively.

Let v be a henselian valuation of arbitrary rank of a field K and v be the prolongation of v to the algebraic closure K ˜ of K with value group G ˜ . In 2008, Ron Brown gave a class P of monic irreducible polynomials over K such that to each g ( x ) belonging to P , there corresponds a smallest constant λ g belonging to G ˜ (referred to as Brown’s constant) with the property that whenever v ( g ( β ) ) is more than λ g with K ( β ) a tamely ramified extension of ( K , v ) , then K ( β ) contains a root of g ( x ) . In this paper, we determine explicitly this constant besides giving an important property of λ g without assuming that K ( β ) / K is tamely ramified.