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In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,0, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties...
The asymptotic behaviour of universal fuzzy measures is investigated in the present paper. For each universal fuzzy measure a class of fuzzy measures preserving some natural properties is defined by means of convergence with respect to ultrafilters.
We prove that, under CH, for each Boolean algebra A of cardinality at most the continuum there is an embedding of A into P(ω)/fin such that each automorphism of A can be extended to an automorphism of P(ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.
We investigate Banach space automorphisms focusing on the possibility of representing their fragments of the form
for A,B ⊆ ℕ infinite by means of linear operators from into , infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for others give...
We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space such that has no uncountable (semi)biorthogonal sequence where ’s are atomic measures with supports consisting of at most 2n-1 points of , but has biorthogonal systems where ’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is...
An abstract form of modus ponens in a Boolean algebra was suggested in [1]. In this paper we use the general theory of Boolean equations (see e.g. [2]) to obtain a further generalization. For a similar research on Boolean deduction theorems see [3].
We study the Borel reducibility of Borel equivalence relations on the generalized Baire space for an uncountable κ with . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.
(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”.
(ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”.
(iii) Under P(ω) the following two statements are equivalent:
(a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter.
(b) Every Boolean algebra of...
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