MRa-MATH: Lattice Multiresolution Framework

• MRa-MATH is a framework that unifies multiresolution analysis and wavelet techniques with lattice-theoretic constructs, using primorial lattices to mirror nested subspaces.
• The framework introduces novel reduction, difference, and projection operators that serve as analogues to classical scaling, detail extraction, and projection in signal processing.
• MRa-MATH enables practical applications in logic, fuzzy probability, and genomic sequence analysis by offering multiscale filtering and structured information decomposition.

MRa‑MATH is a framework that unifies multiresolution analysis (MRA) and wavelet subspace techniques with lattice-theoretic constructions, providing a mathematical infrastructure for logic, probability, and symbolic sequence processing at multiple resolutions. The foundational premise is that the traditional hierarchy of subspaces in an MRA or wavelet analysis can be mirrored by nested Boolean or orthocomplemented lattices, called "primorial lattices," with novel reduction, difference, and projection operators serving as analogues to classical scaling, detail, and projection operators. This design enables the translation of multiresolution concepts—such as filtering, localization, and scale separation—into the structural language of logic and nonnumeric signal processing.

1. Primorial Lattices and Hierarchical Subspace Structure

MRa-MATH introduces the concept of a primorial lattice as a generalization of the subspace lattice underlying any MRA construction. In the classical setting, a Boolean lattice $L$ of $2^N$ elements (i.e., the lattice of subsets of an $N$-element set) serves as the base structure. A lattice $L$ is called primorial if:

• $0$ is the least element,
• $L$ is atomic with specified atoms $\{y_0, x_0, x_1, ..., x_n\}$,
• There is a sequence $y_{n+1} = y_n \lor x_n$ for $0 \leq n < N$.

The recursive nature (each $y_{n+1}$ formed by joining the previous resolution's $y_n$ with an atom $x_n$) induces a structure that mirrors the nested subspaces of MRAs. Primorial lattices manifest in various forms depending on the choice of atoms and join rules, and allow for a systematic generation of lattices supporting operations at different granularity or abstraction levels.

2. Lattice Reduction and Difference Operators

Key to MRa-MATH are operators for reducing and comparing lattices, analogous to scaling and detail extraction in classical MRA:

• Reduction operator $R$: Reduces a Boolean lattice of $2^N$ elements to a lower-resolution lattice, e.g., from $L_2^N$ to $L_2^{N-1}$. For instance, a $2^3$-element lattice is reduced to several $2^2$-element lattices, with each capturing coarsened logical or probabilistic relationships.
• Difference operator $\setminus$: For two bounded lattices $L_X$ and $L_Y$ (with a shared bottom and top), the difference lattice $L_X \setminus L_Y$ is the set $X \cup Y \cup \{0, 1\}$ with inherited partial order. This operator enables the construction of "detail" lattices representing elements present at finer but not coarser levels, producing a sequence of orthocomplemented lattices paralleling wavelet detail spaces.

This hierarchy and the accompanying operators induce a decomposition of logical or symbolic information into different "resolution" and "frequency" components, much as classical MRA decomposes and reconstructs numeric signals.

3. Lattice Projections and Multiresolution Operators

MRa-MATH provides several projection operators to map high-resolution logical or symbolic data onto coarser lattices:

• Zero primorial projection: Elements not present in a lower-resolution lattice project to $0$ ("null"), ensuring coarsened logic is well-defined and discards unresolvable detail.
• Sasaki primorial projection: For orthocomplemented lattices, the Sasaki projection $P_x(y) = (x \wedge y) \lor (x' \wedge y)$ projects an element $y$ with respect to $x$, combining with the join operation over relevant sublattices for a "minimal subspace cover."
• Metric primorial projection: Utilizing the height function $h:L \to \mathbb{N}$ as a discrete valuation, this projection finds the nearest neighbor in the target lattice under $d(x, y) = h(x \lor y) - h(x \wedge y)$.

These projections simulate the downsampling and restriction operations in numerical MRA but are structurally adapted to the combinatorics of lattices.

4. Lattice-Valued Probability and Logical Calculi

MRa-MATH extends probability measures to lattice-valued functions, essential for defining and reasoning about uncertainty, membership, or fuzzy logic in non-numeric settings:

Given a lattice $L=(X, \vee, \wedge, ', 0, 1; \leq)$ with a defined negation, a probability function $p: X \to \mathbb{R}$ must satisfy:

1. $p(0) = 0$, $p(1) = 1$ (normalization),
2. $x \leq y \implies p(x) \leq p(y)$ (monotonicity),
3. For $x \wedge y = 0$, $p(x \vee y) = p(x) + p(y)$ (orthogonal additivity).

On Boolean lattices, the familiar relation $p(x \vee y) = p(x) + p(y) - p(x \wedge y)$ is recovered. For non-distributive or orthomodular lattices, additivity is limited to orthogonal events, generalizing Kolmogorov’s axioms to broader logical frameworks. This enables probabilistic reasoning and information-theoretic analysis in logics for quantum or symbolic probability spaces.

5. Applications: Logic, Probability, and Symbolic Sequence Processing

The MRa-MATH framework enables multiresolution and multiscale analysis for a variety of data types:

Application Area	MRa-MATH Structure	Example
Logic	Boolean/primorial lattices	Propositions at multiple resolutions, projection between coarse/fine logics
Probability	Lattice-valued measures	Multiscale event spaces, reasoning in non-Boolean or fuzzy contexts
Symbolic Sequences	Projected lattices	Genomic signal processing, logical filtering of sequence motifs

• Logic Analysis: Propositions in high-resolution Boolean logic can be projected to coarser logics using primorial lattices, supporting multiscale logical inference.
• Fuzzy Logic: Membership functions decomposed into monotonic/non-monotonic parts at different resolutions via primorial reduction (e.g., extracting robust features from fuzzy sets).
• Probability Analysis: Lattice-valued probability allows for uncertainty quantification in non-distributive event spaces relevant to quantum logic and finite-automaton-based probability.
• Symbolic Sequence Processing: DNA or symbolic sequences encoded as atomic lattice elements can be filtered by projecting onto coarse lattices, akin to low-frequency or "summary" motifs. For instance, projection highlights regions rich in specific nucleobases (AT/CG) or repeats, supporting genomic signal processing.

6. Theoretical and Practical Significance

The primorial lattice MRa-MATH framework bridges numerical, logical, and symbolic processing, offering:

• A canonical generalization of wavelet/MRA ideas to non-numeric domains.
• Theoretical infrastructure for "resolving" logics, probabilities, or symbolics at multiple scales/frequencies.
• Mechanisms for multiscale filtering and adaptive analysis in fields as diverse as digital logic design, fuzzy systems, finite automata, and non-numeric signal processing (e.g., genomics).

Appropriately defined reduction, projection, and difference operators allow for information-preserving (or, by design, lossy) transitions between levels, supporting reasoning, detection, or characterization tasks requiring abstraction across scales.

7. Conclusion

MRa-MATH synthesizes MRA and lattice theory to enable multiresolution analysis for complex, non-numeric data such as logical propositions, probability models, and symbolic sequences. By leveraging primorial lattices, reduction/difference/projection operators, and a generalized notion of lattice-valued probability, it extends the power of classical harmonic and wavelet analysis to broader fields, including fuzzy logic and genomic signal processing. Key mathematical constructs—including explicit formulas for projections, the use of metrics and heights, and generalized probability axioms—anchor the practical and theoretical utility of this methodology for a wide array of multilevel analysis and processing tasks (Greenhoe, 2014).