Let $𝔪$ be an infinite cardinal. We denote by $C_{𝔪}$ the collection of all $𝔪$-representable Boolean algebras. Further, let $C_{𝔪}^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_{𝔪}$. In this paper we prove that $C_{𝔪}^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.

Let B(κ,λ) be the subalgebra of P(κ) generated by ${\left[\kappa \right]}^{\le \lambda }$. It is shown that if B is any homomorphic image of B(κ,λ) then either $\left|B\right|<2^{\lambda }$ or $\left|B\right|={\left|B\right|}^{\lambda }$; moreover, if X is the Stone space of B then either $\left|X\right|\le 2^{2^{\lambda }}$ or $\left|X\right|=\left|B\right|={\left|B\right|}^{\lambda }$. This implies the existence of 0-dimensional compact $T_2$ spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

Rasiowa and Sikorski [5] showed that in any Boolean algebra there is an ultrafilter preserving countably many given infima. In [3] we proved an extension of this fact and gave some applications. Here, besides further remarks, we present some of these results in a more general setting.

The probability $p\left(s\right)$ of the occurrence of an event pertaining to a physical system which is observed in different states $s$ determines a function $p$ from the set $S$ of states of the system to $[0,1]$. The function $p$ is called a numerical event or multidimensional probability. When appropriately structured, sets $P$ of numerical events form so-called algebras of $S$-probabilities. Their main feature is that they are orthomodular partially ordered sets of functions $p$ with an inherent full set of states. A classical...

We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is ℵ₂-projective. Previously it was known that this characterization of openly generated Boolean algebras follows from Axiom R. Since FRP is preserved by c.c.c. generic extension, we conclude in particular that this characterization is consistent with any set-theoretic assertion forcable by a c.c.c. poset starting from a model of FRP. A crucial step...

The reaping number ${𝔯}_{m,n}\left(𝔹\right)$ of a Boolean algebra $𝔹$ is defined as the minimum size of a subset $𝒜\subseteq 𝔹\setminus \left\{𝐎\right\}$ such that for each $m$-partition $𝒫$ of unity, some member of $𝒜$ meets less than $n$ elements of $𝒫$. We show that for each $𝔹$, ${𝔯}_{m,n}\left(𝔹\right)={𝔯}_{\lceil \frac{m}{n-1}\rceil ,2}\left(𝔹\right)$ as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every $k$-branching tree whose maximal nodes are coloured with $\ell$ colours contains an $m$-branching subtree using at most $n$ colours if and only if $\lceil \frac{\ell }{n}\rceil <\lceil \frac{k}{m-1}\rceil$.

Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or $x^{⊲}\in F$. In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G...