L-Mosaics: Algebraic & Quantum Structures

• L-mosaics are hypercompositional algebraic structures defined by multivalued operations, identity elements, and weak associativity, linking algebra with topology and quantum frameworks.
• They generalize bounded join-semilattices and orthomodular lattices through categorical equivalence, as verified by formal methods in Isabelle/HOL.
• L-mosaics find practical applications in quantum logic, quantum knot theory, and computational topology by providing discrete models for complex quantum phenomena.

L-mosaics are hypercompositional algebraic structures with deep connections to order theory, quantum logic, and discrete representations in topology and computational mathematics. The term encompasses both categorical algebraic frameworks—where L-mosaics generalize bounded join-semilattices and orthomodular lattices via multivalued operations—and combinatorial tiling models, notably in knot theory and quantum knot systems. Recent research includes formal verification of theoretical equivalences in Isabelle/HOL and the application of L-mosaic frameworks to quantum logic and discrete invariants.

1. Algebraic Foundations: Multivalued Operations and Hypercomposition

An L-mosaic is defined on a set $A$ equipped with a binary multivalued operation $\oplus$ (sometimes denoted as $[ ]$ or $\ast$), a distinguished neutral element $e \in A$, and typically a reversibility operator $\rho : A \to A$. Key axioms governing L-mosaics include:

• Commutativity: $x \oplus y = y \oplus x$.
• Diagonal/identity: $e, x \in x \oplus x$.
• Weak associativity: $(x \oplus x) \oplus (x \oplus x) = x \oplus x$.
• Uniqueness for joins: For all $x, y$, there exists a unique $z \in x \oplus y$ with $x, y \in z \oplus z$.

The multivalued operation, central to the hypercompositional nature, maps pairs to nonempty subsets: $x \oplus y \subseteq A$, $\forall x, y \in A$. This non-determinism makes L-mosaics suitable for encoding intrinsically "nonclassical" combinatorial or logical phenomena, such as those arising in quantum logics.

2. Equivalence with Bounded Join-Semilattices

The structure of an L-mosaic can be categorically translated into bounded join-semilattices. In a bounded join-semilattice $(L, \vee, \bot)$:

• $\vee$ (join) is idempotent, commutative, associative.
• $\bot$ is the least element.

The Nakano construction provides the mapping:

$$\text{Nak\_mul}(x, y) = \{z \in L \mid x \vee y = x \vee z \text{ and } x \vee y = z \vee y \}$$

Starting from a bounded join-semilattice, Nakano's multivalued operation yields an L-mosaic. Conversely, from an L-mosaic, the minimal diagonal criteria ensure the extraction of a single-valued join $\sqcup$, which satisfies

$(x \sqcup y) \sqcup z = x \sqcup (y \sqcup z)$

and the expected least upper bound properties. Isabelle/HOL formalization rigorously confirms the mutual inverse property of these constructions (Linzi, 24 Sep 2025). The L-mosaic axioms guarantee the existence and associativity of the derived join operation, confirming the categorical equivalence.

3. Dualizability, Ortholattices, and Orthomodularity

A dualizable L-mosaic introduces an involutive duality $T : A \to A$ (with $T^2 = \mathrm{id}$) so that the $T$-dual structure $(A, \oplus_T, T(e))$ with $x \oplus_T y = T(x) \oplus T(y)$ is itself an L-mosaic.

The correspondence between dualizable L-mosaics and ortholattices is established via functorial equivalence (Cangiotti et al., 10 Jan 2025):

• Ortholattices $(L, \wedge, \vee, 0, 1, T)$ satisfy
  • $x \leq y \iff T(y) \leq T(x)$,
  • $T(x \vee y) = T(x) \wedge T(y)$,
  • $T(x \wedge y) = T(x) \vee T(y)$.

Orthomodularity, crucial in quantum logic, is characterized via:

$x \leq y,\quad T(x) \vee y = 1 \implies x = y$

or equivalently,

$y = x \vee (T(x) \wedge y)$

The equivalence confirms that L-mosaics equipped with $T$ encode all the axiomatic structure of orthomodular lattices, suggesting model-theoretic applications in quantum logic.

4. Computational Formalization and AI-Assisted Verification

The formal equivalence between L-mosaics and bounded join-semilattices is fully mechanized in Isabelle/HOL (Linzi, 24 Sep 2025):

• Theories are structured using locales, layering from multivalued magmas to full L-mosaics.
• The Hilbert choice (THE operator) manages the extraction of unique diagonal join elements promised by L-mosaic axioms.
• Functors for transformation in both directions are proven mutually inverse.
• LLMs are integrated as reasoning assistants, suggesting lemmas, guiding proof tactics, and facilitating set-theoretic manipulations. Human inspection remains essential for validation.

This methodology paves the way for mechanized formalization of further algebraic structures, especially those arising in quantum information theory and categorical logic.

5. Applications to Quantum Logic and Quantum Knot Theory

L-mosaics' algebraic framework provides a robust basis for modeling quantum logical propositions, effect algebras, and quantum event lattices:

• Orthomodular lattices and their categorical equivalence with dualizable L-mosaics position the latter as foundational in quantum logic.
• In quantum knot theory, L-mosaics function as combinatorial representations. The dimension of the Hilbert space spanned by mosaic states matches the count of suitably connected mosaics for a grid of fixed size.
• Mosaic models extend to virtual knots and embeddings on alternative surfaces (toric, Klein bottle), impacting discrete topological invariants and computational knot theory.

6. LaTeX Formulas and Technical Summary Table

Structure	Operation / Relation	Key Formula
L-mosaic	Multivalued $\oplus$	$x \oplus y \subseteq A$
Join-semilattice	Single-valued $\vee$	$(x \vee y) \vee z = x \vee (y \vee z)$
Nakano construction	Multivalued $\oplus$ from join $\vee$	$$\text{Nak\_mul}(x,y) = \{z \mid x \vee y = x \vee z = z \vee y\}$$
Dualizable L-mosaic	Involution $T$; $T$-dual $\oplus_T$	$x \oplus_T y = T(x) \oplus T(y)$
Orthomodular lattice	Orthocomplementation $T$; orthomodularity condition	$x \leq y,\ T(x) \vee y = 1 \implies x = y$
Hilbert space dimension	Mosaic count $D_{m,n}$; basis isomorphic to mosaic states	Computational algorithms determine $D_{m,n}$

7. Broader Implications and Future Directions

The equivalence between L-mosaics and bounded join-semilattices enhances the toolkit for formal paper in categorical algebra, order theory, and quantum structures. With AI-assisted proof strategies now validated in formal environments, large-scale mechanization of algebraic theorems becomes feasible. Extensions to other algebraic systems (e.g., effect algebras, generalized orthomodular structures) are plausible. The combinatorial efficiency of L-mosaic representations in quantum knots and their adaptability to virtual and topological settings suggest ongoing significance in both pure mathematics and computational applications.