### ${\forall}_{n}$-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.

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.

We prove that a Boolean algebra is countable iff its subalgebra lattice admits a continuous complementation.

In the present paper we deal with the relations between direct product decompositions of a directed set $L$ and direct product decompositions of intervals of $L$.