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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
Rises for Measuring Local Distributivity in Lattices
Mohammad Abdulla, Tobias Hille, Dominik Dürrschnabel, Gerd Stumm · 2025-06-29 · via math.CO updates on arXiv.org

Distributivity is a well-established and extensively studied notion in lattice theory. In the context of data analysis, particularly within Formal Concept Analysis (FCA), lattices are often observed to exhibit a high degree of distributivity. However, no standardized measure exists to quantify this property. In this paper, we introduce the notion of rises in (concept) lattices as a means to assess distributivity. Rises capture how the number of attributes or objects in covering concepts change within the concept lattice. We show that a lattice is distributive if and only if no non-unit rises occur. Furthermore, we relate rises to the classical notion of meet- and join distributivity. We observe that concept lattices from real-world data are to a high degree join-distributive, but much less meet-distributive. We additionally study how join-distributivity manifests on the level of ordered sets.