Two equivalent criteria for modularity of the lattice of all physical decision effects
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This paper answers the open question 1 of  in the affirmative and, conditionally, the open question 2 of , too. Assuming irreducibility of the orthomodular latticeG of all physical decision effectsE, we shall prove in the first section that modularity ofG implies symmetry of the physical probability function μ. In the second section, we shall consider the filter algebra ℬ(B′) being assumed to possess an involution * such thatT*T=0 impliesT=0. Then this will be proved: If every atomic filterT P is a fixpoint of * and * is, in a restricted manner, norm-preserving on the minimal left ideal ℒ P :=ℬ(B′)T P , thenG is modular.
KeywordsNeural Network Statistical Physic Complex System Nonlinear Dynamics Probability Function
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