4 research outputs found
A simultaneous generalization of independence and disjointness in boolean algebras
We give a definition of some classes of boolean algebras generalizing free
boolean algebras; they satisfy a universal property that certain functions
extend to homomorphisms. We give a combinatorial property of generating sets of
these algebras, which we call n-independent. The properties of these classes
(n-free and omega-free boolean algebras) are investigated. These include
connections to hypergraph theory and cardinal invariants on these algebras.
Related cardinal functions, Ind, which is the supremum of the cardinalities
of n-independent subsets; i_n, the minimum size of a maximal n-independent
subset; and i_omega, the minimum size of an omega-independent subset, are
introduced and investigated. The values of i_n and i_omega on P(omega)/fin are
shown to be independent of ZFC.Comment: Sumbitted to Orde
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Independent Partitions in Boolean Algebras
This dissertation introduces a generalization of the cardinal invariant independence for Boolean algebras, suggested by the proof of the Balcar-Franek Theorem. The objects of study are independent sets of partitions under this new notion of independence. Generalizations of several known results regarding large and small independence are formulated and proved, and counterexamples provided for others. Notably the Balcar-Franek theorem itself is generalized