### ${\forall}_{n}$-theories of Boolean algebras

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In this note we give a characterization of complete atomic Boolean algebras by means of complete atomic lattices. We find that unicity of the representation of the maximum as union of atoms and Lambda-infinite distributivity law are necessary and sufficient conditions for the lattice to be a complete atomic Boolean algebra.

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

We introduce the so-called DN-algebra whose axiomatic system is a common axiomatization of directoids with an antitone involution and the so-called D-quasiring. It generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebras and Boolean rings.

We compare the forcing-related properties of a complete Boolean algebra $\mathbb{B}$ with the properties of the convergences ${\lambda}_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda}_{\mathrm{ls}}$ on $\mathbb{B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda}_{\mathrm{ls}}$ is a topological convergence iff forcing by $\mathbb{B}$ does not produce new reals and that ${\lambda}_{\mathrm{ls}}$ is weakly topological if $\mathbb{B}$ satisfies condition $\left(\hslash \right)$ (implied by the $\U0001d531$-cc). On the other hand, if ${\lambda}_{\mathrm{ls}}$ is a weakly topological convergence, then $\mathbb{B}$ is a ${2}^{\U0001d525}$-cc algebra...

It was proved by Juhász and Weiss that for every ordinal α with $0<\alpha <{\omega}_{2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that ${\kappa}^{<\kappa}=\kappa $ and α is an ordinal such that $0<\alpha <{\kappa}^{++}$, 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 this forcing through all $\alpha <{\kappa}^{++}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic...

We present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.

We show that splitting of elements of an independent family of infinite regular size will produce a full size independent set.

In this paper, we establish a theorem on Möbius inversion over power set lattices which strongly generalizes an early result of Whitney on graph colouring.