We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’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 ${\kappa }^{\kappa }$ for an uncountable κ with ${\kappa }^{<\kappa }=\kappa$. 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 $2^{\left(\omega \right)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{\left(\omega \right)}$ 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...

We partially strengthen a result of Shelah from [Sh] by proving that if $\kappa ={\kappa }^{\omega }$ and $P$ is a CCC partial order with e.g. $\left|P\right|\le {\kappa }^{+\omega }$ (the ${\omega }^{\text{th}}$ successor of $\kappa$) and $\left|P\right|\le 2^{\kappa }$ then $P$ is $\kappa$-linked.

If element $z$ of a lattice effect algebra $(E,\oplus ,\mathbf{0},\mathbf{1})$ is central, then the interval $[\mathbf{0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C\left(E\right)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C\left(E\right)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C\left(E\right)$ is a bifull sublattice...

In set theory without the axiom of choice ($\mathrm{AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC${}^{\mathrm{LO}}$ (AC for linearly ordered families of nonempty sets)—and hence AC${}^{\mathrm{WO}}$ (AC for well-ordered families of nonempty sets)— $\mathrm{DC}(<\kappa )$ (where $\kappa$ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f:X\to \{0,1\}\times X$”, implies the statement “there exists a field $F$ such that every vector...

We prove that for an unbounded metric space $X$, the minimal character $𝗆\chi \left(\check{X}\right)$ of a point of the Higson corona $\check{X}$ of $X$ is equal to $𝔲$ if $X$ has asymptotically isolated balls and to $max\{𝔲,𝔡\}$ otherwise. This implies that under $𝔲<𝔡$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\mathbb{N}}$ if and only if $dim\left(\check{X}\right)=0$ and $𝗆\chi \left(\check{X}\right)=𝔡$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

A standard bridge between automata theory and logic is provided by the notion of characteristic formula. This paper investigates this problem for the class of event-recording automata (ERA), a subclass of timed automata in which clocks are associated with actions and that enjoys very good closure properties. We first study the problem of expressing characteristic formulae for ERA in Event-Recording Logic (ERL ), a logic introduced by Sorea to express event-based timed specifications. We prove that...