arXiv subjects

Joaquim Reizi Higuchi

Publications and source records attributed to Joaquim Reizi Higuchi.

5 recordsLinked to original sources

arXiv2026

A Survey on Lawvere's Fixed-Point Theorem

Lawvere's fixed-point theorem organizes self-reference around a representation hypothesis and a diagonal calculation. This survey develops its set-theoretic, finite-product, and cartesian closed formulations while distinguishing representation on points from equality of whole morphisms. Detailed applications include Cantor's theorem, arithmetic diagonalization, a truth-schema obstruction, an incompleteness consequence with explicit assumptions, the halting problem, and Kleene's recursion theorem. A powerset dcpo supplies a nontrivial positive representation of continuous maps and illustrates the difference between fixed-point existence, a uniform operator, and a least fixed point. Further sections compare parameterized recursion and trace, substructural and relational generalizations, constructive witness data, and topological variants. Throughout, the relevant maps, equality relations, and additional hypotheses are kept explicit. Worked examples use finite tables, Boolean equations, numerical function codes, factorial approximations, and self-output programs. The survey is intended for readers beginning category theory who wish to understand both the common diagonal mechanism and the work required in its applications.

math.GM↗

arXiv2026

A Countable Three-Generated Orthomodular Lattice with No $σ$-Complete Embedding

We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any $σ$-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term $a$ is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term $X$ satisfies $π_aπ_X(t)=a$. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of $a$ to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.

math.LO↗

arXiv2026

The first fatal axiom for weakened sequential products on finite MV-effect algebras: Local obstruction, exact low-rank classification, and the rank-one boundary case

We study total binary operations on effect algebras obtained by truncating the Gudder-Greechie axiom package for a sequential product. The point is not to reprove the known nonexistence of non-Boolean full sequential products on finite chains, but to determine, axiom by axiom, where finite MV-effect algebras first fail. We prove two structural facts valid on every effect algebra. First, the operation σ_E(a,b) = 0 if a=0, and b if a \neq 0, satisfies (S1)-(S3), so (S3) is never fatal by itself. Second, any operation satisfying (S1)-(S4) already has the right-unit property a \circ 1 = a, even without (S5). From this we derive a local obstruction theorem: if an effect algebra contains an atom of finite isotropic index at least 2, then it admits no (S1)-(S4) operation. Consequently, a finite MV-effect algebra admits such an operation if and only if it is Boolean. In this precise sense, (S4) is the first fatal axiom on finite MV-effect algebras. On the constructive side, let E_u = [0,u] \subseteq Z^r be the simplicial interval representation of a finite MV-effect algebra. We show that additive maps E_u \to E_v are exactly the restrictions of positive group homomorphisms Z^r \to Z^s, equivalently maps x \mapsto Mx given by nonnegative integer matrices with Mu \le v. This yields a complete classification of (S1)+(S2) operations by row-wise subunital matrices. We then solve the first genuinely higher-rank (S1)-(S3) problem: on the rank-two Boolean algebra B_2 = E_{(1,1)} \cong C_1^2, all such operations are classified and there are exactly 34. Thus the finite-chain collapse at (S3) is a rank-one boundary phenomenon, whereas on finite MV-effect algebras the sharp threshold for nonexistence occurs exactly at (S4).

math.LO↗

arXiv2026

Three Fixed-Dimension Satisfiability Semantics for Quantum Logic: Implications and an Explicit Separator

We compare three satisfiability notions for propositional formulas in the language {not, and, or} over a fixed finite-dimensional Hilbert space H=F^d with F in {R, C}. The first is the standard Hilbert-lattice semantics on the subspace lattice L(H), where meet and join are total operations. The second is a global commuting-projector semantics, where all atoms occurring in the formula are interpreted by a single pairwise-commuting projector family. The third is a local partial-Boolean semantics, where binary connectives are defined only on commeasurable pairs and definedness is checked nodewise along the parse tree. We prove, for every fixed d >= 1, Sat_COM^d(phi) implies Sat_PBA^d(phi) implies Sat_STD^d(phi) for every formula phi. We then exhibit the explicit formula SEP-1 := (p and (q or r)) and not((p and q) or (p and r)) which is satisfiable in the standard semantics for every d >= 2, but unsatisfiable under both the global commuting and the partial-Boolean semantics. Consequently, for every d >= 2, the satisfiability classes satisfy SAT_COM^d subseteq SAT_PBA^d subset SAT_STD^d and SAT_COM^d subset SAT_STD^d, while the exact relation between SAT_COM^d and SAT_PBA^d remains open. The point of the paper is semantic comparison, not a new feasibility reduction or a generic translation theorem.

cs.LO↗

arXiv2025

The Strict 2-Category Structure of Distorted Monoidal Categories

This paper introduces the concept of distorted monoidal categories, a generalization of monoidal and braided monoidal categories that supports non-reversible and direction-sensitive tensor structures. Unlike the classical setting, where the braiding symmetry is required to be invertible, distorted monoidal categories admit non-invertible binary distortions and unit distortions while preserving coherent tensorial reasoning. We show that these structures naturally assemble into a strict 2-category whose composition and interchange laws hold on the nose, not merely up to isomorphism. Beyond the abstract 2-monad justification, our contribution is a fully constructive and type-safe calculus that enables formal reasoning about non-invertible interchange. We provide explicit construction schemes for such distortions, including idempotent twists of classical braidings and graded unit distortions arising from characters on monoidal gradings. This framework extends the expressive power of monoidal categories to model irreversible, resource-sensitive, and direction-dependent processes --such as those in directed homotopy theory, categorical quantum mechanics, and non-symmetric operadic structures--while remaining amenable to mechanization and formal verification.

math.CT↗