In a previous paper, we introduced the notion of Boolean-like algebra as a generalisation of Boolean algebras to an arbitrary similarity type. In a nutshell, a double-pointed algebra A with constants 0, 1 is Booolean-like in case for all a ∈ A the congruences θ (a, 0) and θ (a, 1) are compementary factor congruences of A. We also introduced the weaker notion of semi-Boolean-like algebra, showin...

Von Neumann conjectured that the countable chain condition and the weak (ω, ω)-distributive law characterize measurable algebras among Boolean σalgebras [Mau]. Consistent counter-examples have been obtained by Maharam [Mah], Jensen [J], Glówczyński [Gl], and Veličković [V]. However, whether von Neumann’s proposed characterization of measurable algebras fails within ZFC remains an open problem. ...

It was proved by Juhász and Weiss that for every ordinal α with 0 < α < ω2 there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that κ = κ and α is an ordinal such that 0 < α < κ++, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating...

The theory of Boolean algebras can be fruitfully applied to judgment aggregation: Assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent com...

It is a well known problem of Von Neumann whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar–Jech–Pazák and Velickovic that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra...

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B0 × B1) = max{irr(B0), irr(B1)}. We prove consistency of the statement “there is a Boolean algebra B such that irr(B) < s(B ~ B)” and we force a superatomic Boolean algebra B∗ such that s(B∗) = inc(B∗) = κ, irr(B∗) ...

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many such endomorphisms in some reasonable ways. We prove that in any cardinality λ = λ0 there is a Boolean algebra with no other endomorphisms. For this we use the...

In this paper, we prove that every Pre A∗-algebra A with 1 is isomorphic to the Pre A∗-algebra of all global sections of a sheaf of indecomposable Pre A∗-algebra over Boolean space and that sheaf of indecomposable Pre A∗-algebra with 1 over Boolean space is isomorphic to the sheaf obtained from the Pre A∗-algebra of all sections of the sheaf. Copyright c © 2011 Yang’s Scientific Research Instit...

Extended Abstract In this Note I prove a representation theorem for Peirce algebras introduced in Brink, Britz and Schmidt (1994b, 1994a). I show that the class of full Peirce algebras is characterised by the class of complete and atomic Peirce algebras in which the set of relational atoms is restricted by two conditions. One requires simplicity. The other requires that each relational element ...