The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations.
A subalgebra B of the direct product of Boolean algebras is finitely closed if it contains along with any element f any other member of the product differing at most at finitely many places from f. Given such a B, let B* be the set of all members of B which are nonzero at each coordinate. The generalized free product corresponding to B is the subalgebra of the regular open algebra with the poset topology on B* generated by the natural basic open sets. Properties of this product are developed....
We show that, consistently, for some regular cardinals θ <λ, there exists a Boolean algebra 𝔹 such that |𝔹| = λ⁺ and for every subalgebra 𝔹'⊆ 𝔹 of size λ⁺ we have Depth(𝔹') = θ.
The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space...
The variety of basic algebras is closed under formation of horizontal sums. We characterize when a given basic algebra is a horizontal sum of chains, MV-algebras or Boolean algebras.
We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komjáth P., Boolean algebras in which every chain and antichain is countable, Fund. Math. 111 (1981), 125–133, Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117, and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, such that , answering questions raised by Monk...