poset theory

This work addresses the algebraic classification and tractability boundary of constraint satisfaction problems (CSPs). Focusing on conservative CSP templates, we develop a unified theoretical framework linking absorptive algebras to local consistency solvability—the first systematic characterization of their deep interplay. Using subpower-finite algebraic modeling, absorptive subalgebra analysis, and polynomial-time reductions, we rigorously establish a complexity dichotomy theorem for conservative CSPs: every template is either solvable in polynomial time via local consistency algorithms or NP-complete. Our result provides a complete tractability criterion for conservative CSPs and strengthens the expressive power of algebraic methods in CSP classification. Moreover, it furnishes a rigorous theoretical foundation and decision framework for consistency-based solving approaches.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

Counting minimum-cost solutions to NP-hard combinatorial optimization problems—such as Connected Dominating Set and Constraint Satisfaction—is computationally challenging due to the interplay between cost minimization and solution enumeration. Method: We propose a unified dynamic programming framework grounded in semiring algebra, centered on a novel Δ-product operation that relaxes the idempotence requirement of classical semirings, enabling efficient counting of minimum-cost solutions over non-idempotent semirings. The framework is parameterized by treewidth and clique-width to ensure tractability on structured inputs. Contribution/Results: We establish fixed-parameter tractable (FPT) enumeration for minimum-cost solutions under bounded treewidth or clique-width, proving polynomial-time solvability of the counting problem in these parameters. Our approach significantly extends classical DP’s expressiveness, unifying cost optimization and solution counting within a single algebraic model—thereby enabling rigorous analysis and efficient computation for previously intractable enumeration tasks.

This paper addresses the lack of a unified framework for rewriting theory in strict higher-dimensional categories. Methodologically, it reconstructs algebraic rewriting theory around the central notion of *polygraphs*, embedding them systematically into the setting of strict $n$-categories and homotopical algebra, thereby developing a comprehensive $n$-polygraph theory; notably, it proves—within the folk model structure—that $n$-polygraphs are cofibrant objects, the first such result. The main contributions are threefold: (i) a deep unification of polygraphs with strict higher-dimensional categories and homotopical algebra; (ii) a systematic generalization of low-dimensional confluence results to arbitrary dimension; and (iii) a universal criterion and algorithmic procedure for verifying coherence of higher-dimensional algebraic structures. These advances provide novel structural tools for higher algebra, type theory, and formal verification.

A fundamental information asymmetry exists between measurement-based mathematical representations and optimization-principle-based physical representations of experimental setups: certain partial orders require infinite information for physical realization, yet admit finite-information mathematical descriptions. Method: We introduce the “Debreu dimension”—a novel dimension concept unifying geometric intuition with order-theoretic rigor—thereby bridging the semantic gap between Dushnik–Miller dimension and classical geometric dimension. Under countability assumptions, we develop an explicit, geometry-guided constructive framework, integrating order theory, real-valued monotone function analysis, and set-theoretic tools. Contribution/Results: This work establishes a comprehensive dimensional classification scheme for preordered spaces, significantly enhancing both the geometric unification of partially ordered sets and the classification accuracy of real-valued monotone representations.

This paper addresses the structured lifting problem by proposing the first systematic abstract framework that uniformly characterizes its mathematical structure and computational semantics. Methodologically, it integrates lifting property analysis from category theory, structured abstract modeling, and type-theoretic semantic techniques. The work establishes, for the first time, a rigorous proof that lifting solutions within this framework are universally existent, closed under relevant operations, and unique. These results not only uncover fundamental regularities underlying structured lifting but—crucially—bridge categorical lifting with type-theoretic computation. In particular, they provide a rigorous semantic foundation for axiomatizing computational rules in cubical type theory, thereby advancing formalization and computability research in higher-order type theory.

This study investigates the computational complexity of constraint satisfaction problem (CSP) templates induced by semilattice Maltsev block (SMB) algebras—structures that endow each element of a semilattice with a Maltsev algebra. Employing tools from universal algebra, CSP complexity theory, and the analysis of Maltsev structures, the authors provide the first complete proof that every SMB algebra gives rise to a tractable CSP template. This result not only establishes global tractability for CSPs in the SMB setting but also reveals a deep unification of two classical dichotomy theorem proofs within this framework, thereby generalizing and integrating prior work by Bulatov and others.

This work proposes an abstract and unified categorical semantics for Martin-Löf type theory with explicit universe polymorphism. By employing generalized algebraic theories, it constructs two classes of categorical structures—families of categories equipped with an external tower of universes and families indexed by universe levels—corresponding respectively to two variants of the type theory. The higher-order semantic structure of each is revealed through their initial models. This study is the first to incorporate explicit universe polymorphism into the framework of generalized algebraic theories, thereby abstracting away syntactic details and highlighting its intrinsic categorical nature. The results not only demonstrate the applicability of this framework to sophisticated type systems but also provide a crucial instance supporting Voevodsky’s initiality conjecture.

This paper addresses the axiomatization of sorting functions over sets lacking a predefined total order. We propose an abstract sorting model based on sections (right inverses) of free commutative monoids, defining sorting functions as list operations satisfying specific algebraic axioms—thereby eliminating reliance on a priori total orders. Using functorial signatures and universal algebraic techniques, we formally construct free and commutative monoids within single-sorted type theory, with all proofs mechanized in Cubical Agda. Our main contributions are threefold: (1) the first rigorous bijection between decidable total orders and correct sorting functions; (2) the identification of sorting as fundamentally equivalent to constructing algebraic sections that induce total orders; and (3) a novel, comparison-free mathematical foundation for sorting algorithms grounded in monoid theory and constructive algebra. This framework enables reasoning about sorting independently of element-wise comparisons, offering new insights into its computational and structural essence.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.