Notes on sums of Boolean algebras and logics

Boolean Methods and Pták's Sum

“…Then by transfer, we have, The discrepancy of the Corollary 3.3 with Theorem 3.3 in [7], seems to originate in the fact that in [7] everything is conducted in an ordinary external way, see also Example 4.2. However most of the results in [7], [8] can be taken internally by transfer.…”

“…In this section, for notation and concepts on Quantum Logics see, [12] and for Ptak's sum see [11,7]. The main result of this section is to establish for the first time, a formal connection between Ptak's sum and bounded Boolean powers.…”

“…The intended interpretation by Pták himself and by others who work on it [7], [8], is that the Pták sum is the combination of a Boolean experiment B, with a non-Boolean one L. Since also Β and L are embedded in Β ® L, then we try to study Β ® L in terms of the characteristic of both Β and L. This constitutes an external absolute interpretation that is with respect to ordinary ZFC set theory.…”

“…This, so called Pták's sum, have been investigated by many researches, see e.g. [7,8]. Also in [11] an open question was posed : "Let Κ, M be logics of two independent experiments and let we consider these two experiments as a single one.…”

The Homomorphism on the Ptak's Sum

“…Summing of a Boolean algebra and quantum logic has been defined by P. Ptak [2] and studied in Janis and Riecanova [4] and Janis [3]. First we introduce the definition.…”