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The linear subspaces of a multiresolution analysis (MRA) and the linearsubspaces of the wavelet analysis induced by the MRA, together with the setinclusion relation, form a very special lattice of subspaces which herein iscalled a "primorial lattice". This paper introduces an operator R that extractsa set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice.Used recursively, a sequence of Boolean lattices with decreasing order isgenerated---a structure that is similar to an MRA. A second operator, which isa special case of a "difference operator", is introduced that operates onconsecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence oforthocomplemented lattices. These two sequences, together with the subsetordering relation, form a primorial lattice P. A logic or probabilityconstructed on a Boolean lattice L_2^N likewise induces a primorial lattice P.Such a logic or probability can then be rendered at N different "resolutions"by selecting any one of the N Boolean lattices in P and at N different"frequencies" by selecting any of the N different orthocomplemented lattices inP. Furthermore, P can be used for symbolic sequence analysis by projectingsequences of symbols onto the sublattices in P using one of three latticeprojectors introduced. P can be used for symbolic sequence processing byjudicious rejection and selection of projected sequences. Examples of symbolicsequences include sequences of logic values, sequences of probabilistic events,and genomic sequences (as used in "genomic signal processing").

MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing