In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. We also look at some specific partial orders in the basic Cohen model. Lately we have been thinking about forcing over models of set theory which do not satisfy the axiom of choice (see Miller [8, 9]). One of the first uses of the axiom of choice in forcing is: Maximum Principle p ∃x θ(x) iff ther...

Let S(RG) be a normed Sylow p-subgroup in a group ring RG of an abelian group G with p-component Gp and a p-basic subgroup B over a commutative unitary ring R with prime characteristic p. The first central result is that 1 + I(RG;Bp) + I(R(p)G;G) is basic in S(RG) and B[1 + I(RG;Bp) + I(R(p )G;G)] is p-basic in V (RG), and [1 + I(RG;Bp) + I(R(p )G;G)]Gp/Gp is basic in S(RG)/Gp and [1 + I(RG;Bp)...

The authors give a consistent affirmative response to a question of Juhász, Soukup and Szentmiklóssy: If GCH fails, there are (many) extraresolvable, not maximally resolvable Tychonoff spaces. They show also in ZFC that for ω < λ κ , no maximal λ-independent family of λ-partitions of κ is ω-resolvable. In topological language, that theorem translates to this: A dense, ω-resolvable subset of a s...

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ k[X ] such that f | Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if (1) there is a nonempty open subset U ⊆ X consisting of clo...

In this paper we wish to prove that under certain conditions the Fourier coefficients of the Eisenstein series for an arithmetic group acting on a tube domain are all rational numbers. Let G be a connected, simply-connected, semisimple, and almost Q-simple linear algebraic group defined over the rational number field Q. Let R be the real number field. Then GR is connected, and we assume that if...

Let Γg,b denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus g with b punctures. For a group G let Φf (G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φf (G) is nilpotent when G is a finitely generated subgroup of Γg,b, in this paper we compute Φf (G) for certain subgroups of Γg,b...

Introduction. Two nonempty subsets of a topological space are said to be separated if neither intersects the closure of the other. A set is connected if it is not the union of two separated sets. A connected set I is indecomposable if it is not the union of two connected sets, neither of which is dense in I. In [l], Swingle raised the following question: does there exist, in the plane, an indec...

The notation and terminology used in this paper are introduced in the following articles: [20], [21], [1], [3], [22], [4], [5], [19], [10], [18], [7], [17], [11], [2], [8], [9], [16], [13], [14], [15], [6], [23], and [12]. In this paper p1, p2 are points of E 2 T, C is a simple closed curve, and P is a subset of E T. Let n be a natural number, let A be a subset of En T, and let a, b be points o...

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R ·S where R is the solvable radical of G. We also prove that any semisimple subalgebra of t...

The aim of this paper is to study a certain type of chamber system. For notation and definitions concerning chamber systems see [22]. For the convenience of the reader, we recall two definitions which are of importance in this paper. Let % = (% ($,•),-€/) be a chamber system over / and c € %. By Ay(c) with / c /, we denote the element in £%y containing c (Ay(c) is a chamber system over / ) . Le...