Quantales carrying ortholattice structure

Abstract: This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and orthocomplementation requirements. Our main contribution is a study of Girard posets with inversions, providing a characterization theorem where a unital residuated poset is Girard if and only if it admits an inversion satisfying specific adjointness conditions. We prove that any complemented lattice admitting an integral residuated structure must be Boolean, which motivates our search for orthomodular examples in the non-integral case. We answer this by demonstrating that the lattice $C(\mathbb{R}^n)$ of closed subspaces of $n$-dimensional real coordinate space carries both an orthomodular and a commutative Girard quantale structure. This construction provides a concrete non-Boolean framework unifying quantum-logical and many-valued logical reasoning.

• The paper characterizes Girard posets through order-reversing involutions and a cyclic adjointness condition, providing a practical test for identifying compatible residuated structures.
• The paper proves every complemented integral residuated lattice—and every complemented integral quantale—is Boolean, showing that non-Boolean quantum behavior requires multiplication distinct from lattice meet.
• The paper constructs non-Boolean commutative Girard quantales on closed subspaces of ℝⁿ, where span-of-products multiplication makes orthocomplementation coincide with linear negation, while highlighting open questions for complex spaces and proof systems.

This paper investigates whether the algebraic semantics of many-valued logic—residuated posets and quantales—can coexist with the orthocomplementation structure of quantum logic. The authors, Botur, Kruml, and Paseka, establish both an impossibility result (complemented integral residuated lattices must be Boolean) and a constructive result (the lattice $C(\mathbb{R}^n)$ of closed subspaces of $\mathbb{R}^n$ is simultaneously an orthomodular lattice and a commutative Girard quantale whose orthocomplement coincides with linear negation) (2602.21898).

Background and motivation

Two algebraic traditions frame the work. On one side, residuated lattices and quantales provide semantics for substructural logics; a quantale is a complete lattice with an associative multiplication distributing over arbitrary joins, and every quantale is residuated via $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$. A Girard quantale carries a cyclic dualizing element $d$, yielding a "linear negation" involution $x^{-}= x \rightarrow d$ that algebraically mirrors Girard's treatment of negation in linear logic. On the other side, orthomodular lattices—the canonical example being closed subspaces $C(H)$ of a Hilbert space under set intersection and orthogonal complement—model quantum propositions. The paper asks whether these structures can be unified: can a non-Boolean ortholattice also carry a genuine residuated structure?

The central technical device is the theory of Girard posets with inversions. An inversion on a poset is an involutive antitone self-map; in a Girard poset, the induced linear negation is precisely such an inversion, so the question becomes when an inversion witnesses the Girard property.

Characterizing Girard posets via inversions

The paper first proves two structural facts about Girard posets equipped with an inversion $^{-}$:

• The cyclic dualizing element is uniquely determined as $d = \bigvee_{x}(x \odot x^{-})$.
• A Girard poset with inversion is a Boolean algebra if and only if it is idempotent with dualizing element $d = 0$. This follows since idempotency forces $\odot = \wedge$, making the lattice distributive.

The main characterization theorem states three equivalent conditions for a unital residuated poset $\mathbb{R}^n$0 with unit $\mathbb{R}^n$1: (1) $\mathbb{R}^n$2 is a Girard poset; (2) $\mathbb{R}^n$3 admits an inversion satisfying $\mathbb{R}^n$4; (3) $\mathbb{R}^n$5 admits an inversion satisfying the cyclic adjointness scheme $\mathbb{R}^n$6 for all $\mathbb{R}^n$7. Condition (3) is notable because it characterizes Girardness without mentioning the dualizing element at all—it is exactly this criterion the authors later invoke to verify the Girard property of their spatial examples (2602.21898).

Integral residuation forces Booleanity

The negative half of the paper sharpens classical results of Ward–Dilworth. For any complemented integral residuated lattice, integrality plus join-distributivity of multiplication yields idempotency ($\mathbb{R}^n$8), then $\mathbb{R}^n$9, hence distributivity and Booleanity. Consequently, a complemented lattice is an integral residuated lattice if and only if it is a Boolean algebra—a strengthening of Ward and Dilworth's Theorem 7.31, which required commutativity; the present proof removes that hypothesis and proceeds independently. At the quantale level this means a complemented lattice is an integral quantale iff it is a complete Boolean algebra.

Further, in any unital residuated lattice carrying an orthomodular complementation, the downset $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$0 of the unit is itself a Boolean algebra (via the standard fact that principal downsets of orthomodular lattices are orthomodular), and all its elements are mutually compatible in $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$1, so $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$2 sits inside a single Boolean block. This constrains how far any non-Boolean unital example can deviate from distributive behavior near the unit.

Orthomodular commutative Girard quantales on $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$3

The constructive centerpiece answers affirmatively whether non-Boolean orthomodular unital commutative quantales exist. Viewing $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$4 as a ring under pointwise addition and multiplication, define on $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$5:

$$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$6

the linear span of pointwise products. This operation is associative, commutative, distributes over joins, and has unit $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$7. Taking the dualizing element $$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$8—the subspace of vectors whose coordinate sum vanishes—the key computation shows

$$y \rightarrow z = \bigvee\{t \mid t \odot y \le z\}$$9

which is condition (3) of the characterization theorem. Hence $d$0 is a commutative Girard quantale in which the orthocomplement $d$1 coincides with the linear negation $d$2. Since $d$3 is non-distributive for $d$4... actually, for finite-dimensional $d$5 with $d$6 the subspace lattice is non-Boolean, so the construction delivers a concrete family of non-Boolean structures where quantum complementation and Girard residuation coincide on the nose. Note the multiplication here is not meet: it is strictly larger than intersection in general, which is what circumvents the Booleanity barrier.

The paper also recalls prior results showing $d$7 is simultaneously orthomodular and a non-commutative involutive Girard quantale for $d$8, and develops infinite-dimensional examples: the Schatten-class spectra $d$9 form a chain of quantale homomorphisms indexed by $x^{-}= x \rightarrow d$0, with the norm-exponent arithmetic $x^{-}= x \rightarrow d$1 mirroring the Łukasiewicz product $x^{-}= x \rightarrow d$2 on $x^{-}= x \rightarrow d$3. These assemble into a single involutive Girard quantale $x^{-}= x \rightarrow d$4 of compatible families, on which the trace-based orthogonality induces an orthomodular lattice structure; notably, the argument shows every bounded operator decomposes uniquely as $x^{-}= x \rightarrow d$5, as in finite dimensions. The spectrum of $x^{-}= x \rightarrow d$6 itself, however, fails to be Girard.

Limitations and open questions

Several caveats delimit the results. The positive theorem concerns real coordinate spaces only; the authors explicitly leave open whether $x^{-}= x \rightarrow d$7 admits a compatible quantale structure, and if not, what obstruction prevents it. The infinite-dimensional examples rely on the Schatten-scale framework and the assumption that $x^{-}= x \rightarrow d$8 is norm-closed within $x^{-}= x \rightarrow d$9 for the decomposition argument. While $C(H)$0 shows non-commutative orthomodular Girard quantales exist, no systematic account of that phenomenon is given. Finally, no logic is attached to the new structures: the question of a sound and complete proof system for orthomodular Girard quantales—one combining quantum logic and linear logic—remains open.

Conclusion

The paper delineates a precise boundary for unifying residuated and orthocomplemented structures: integrality collapses complemented lattices to Boolean algebras, while the unital but non-integral setting supports genuinely non-Boolean examples, most concretely $C(H)$1 with span-of-products multiplication, where orthocomplement equals linear negation. The inversion-based characterization of Girard posets provides a reusable verification tool, and the operator-algebraic constructions connect the framework to Schatten ideals and Łukasiewicz arithmetic. The open status of the complex case and the missing logical calculus are the natural next targets for this line of inquiry (2602.21898).