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Given a lattice path \(\nu\), the \(\nu\)-Tamari lattice and the \(\nu\)-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above \(\nu\). In this paper, we introduce a more general family of lattices, called alt \(\nu\)-Tamari lattices, which contains these two examples as particular cases. Unexpectedly, we show that all these lattices have the same number of linear intervals.Mathematics Subject Classifications: 06A07, 06B05, 05A19Keywords: Lattices, intervals, Tamari

Lattice structures are an important class of architected cellular solids and structures with high potential for multifunctional and lightweight applications. Novel technologies such as additive manufacturing have vastly extended the design freedom to develop such architectures. In this work, a reliable theoretical model for optimizing unit cell design against buckling is developed for two different cell architectures: pyramidal and tetrahedral. The model's accuracy was evaluated through extensive finite element analysis and compared to existing methods available in the literature. [ABSTRACT FROM AUTHOR]

Cubic lattices based on tetrakaidecahedron cells with re‐entrant ribs are designed and fabricated via additive manufacturing. Poisson's ratio decreased to a negative value as the re‐entrant angle was made smaller. Poisson's ratio decreased as ribs were made more slender. Lattices with convex tetrakaidecahedron (truncated octahedron) cells had a Poisson's ratio near 0.5 in agreement with theory. Rib slenderness had minimal effect on Poisson's ratio in contrast to the re‐entrant lattices. [ABSTRACT FROM AUTHOR]

Let L be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in L for certain subgroups of the orthogonal group O (L). We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms. [ABSTRACT FROM AUTHOR]

3D printing provides the potential to enhance mechanical properties by fabricating complex structures with diverse materials; however, most high‐resolution 3D printing techniques require custom printers to incorporate multiple materials and/or result in poor material interfacial bonding. Here, energy absorption properties are enhanced with 3D lattice structures fabricated via vat photopolymerization comprising multiple materials forming a gradient‐interpenetrating polymer network (gradient‐IPN). The gradient‐IPN is incorporated by swelling the 3D printed elastomeric lattice in a photoresin that yields a stiff shell‐soft core structure. This straightforward post‐3D printing technique delivers an unprecedented degree of structural property customization through polymer gradients in lattice struts with shells of tunable stiffness and flexible elastomeric cores to achieve a broad continuum spectrum of mechanical properties within one simple system. The gradient aids in the distribution of stress and limits fracture between materials typically observed in multimaterial lattices. The gradient‐IPN lattices are fully recoverable and exhibit over 4 to 33 times higher toughness after compression, compared to copolymer (same composition as the gradient‐IPN) or purely elastomeric lattices, respectively. This highly versatile approach to modifying 3D printed lattices yields the unique combination of load bearing capabilities with viscoelasticity desirable for high performance materials in impact protection. [ABSTRACT FROM AUTHOR]

The category STROPm of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory STROP of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers. [ABSTRACT FROM AUTHOR]

Let O be a Dedekind domain whose field of fractions K is a global field. Let A be a finite-dimensional separable K -algebra and let Λ be an O -order in A. Suppose that X is a Λ-lattice such that K ⊗ O X is free of some finite rank n over A. Then X contains a (non-unique) free Λ-sublattice of rank n. The main result of the present article is to show there exists such a sublattice Y such that the generalised module index [ X : Y ] O has explicit upper bounds with respect to division that are independent of X and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties. [ABSTRACT FROM AUTHOR]

We prove a theorem that generalizes Schmidt's Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace Theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows. [ABSTRACT FROM AUTHOR]

A novel type of lightweight and high-performance, collinear polymer lattices is presented in which the concept of stayed slender columns is exploited with the aid of material extrusion additive manufacturing (MEX). The stays, preventing lower order buckling, are additively manufactured using the printing strategy bridging. Through conducting experimental test series on representative elements and two-dimensional lattices, it is demonstrated that the 3D printed stayed column lattices exhibit significantly improved compressive strength in comparison with conventional collinear lattices. The potential of introducing deliberate geometric imperfections to affect the structural behaviour is furthermore outlined in the current work. [ABSTRACT FROM AUTHOR]

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of solvable unimodular Lie algebras up to dimension 5 which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new attack on NTRU-HPS cryptosystem using lattice theory and Babai's Nearest Plane Algorithm. This attack generalizes the classic CVP attack on NTRU. We present numerical data in support of the validity of our result. [ABSTRACT FROM AUTHOR]

Let K be a number field that is Galois over Q with unit group of rank d. We obtain an upper bound on the number of facets of the reduction domain (and therefore the fundamental domain) of K of O (((d + 1) ρ ∞ (Λ K) + log ⁡ (d 2)) d (d + 1) !) , where ρ ∞ (Λ K) is the covering radius of the log-unit lattice in the infinity norm. As a side-result, we also show that the Weil height of the shortest Pisot unit in the number field can be no greater than γ [ K : Q ] ((d + 1) ρ ∞ (Λ K) + d ϵ) where γ = 1 if K is totally real or 2 otherwise and ϵ > 0 is some arbitrarily small constant. [ABSTRACT FROM AUTHOR]

Multi-prime RSA (MPRSA) is an extended version of RSA in which the modulus is the product of three or more distinct prime numbers. As with the RSA, this variant is also scrutinized for vulnerabilities. The main goal of the attacks on MPRSA is to test the strength of MPRSA compared to that of RSA to determine whether MPRSA can be used in place of RSA. We apply two famous attacks on RSA to MPRSA by constructing lattices using the Jochemsz and May approach. In comparison to RSA, MPRSA is less vulnerable to both attacks. [ABSTRACT FROM AUTHOR]

In this paper, the notions of an L-sublattice, L-ideal, and L-dual ideal of an L-lattice are introduced. The concepts of an L-maximal ideal and an L-prime ideal in an L-lattice are also defined. The present paper also gives the precise structures of an L-sublattice, L-ideal (dual ideal) generated by an L-subset of an L-lattice. Finally, it is proved that under certain conditions, an L-maximal ideal in an L-lattice is L-prime. It is important to note that, in our studies, the evaluation lattice changes from [0,1] to a lattice L and the parent structure shifts from an ordinary lattice to an L-lattice.

Essentially all plasma membrane proteins are glycosylated, and their activity is regulated by tuning their cell surface dynamics. This is achieved by glycan-binding proteins of the galectin family that either retain glycoproteins within lattices or drive their endocytic uptake via the clathrin-independent glycolipid-lectin (GL-Lect) mechanism. Here, we have used immunofluorescence-based assays to analyze how lattice and GL-Lect mechanisms affect the internalization of the cell adhesion and migration glycoprotein α5β1 integrin. In retinal pigment epithelial (RPE-1) cells, internalized α5β1 integrin is found in small peripheral endosomes under unperturbed conditions. Pharmacological compounds were used to competitively inhibit one of the galectin family members, galectin-3 (Gal3), or to inhibit the expression of glycosphingolipids, both of which are the fabric of the GL-Lect mechanism. We found that under acute inhibition conditions, endocytic uptake of α5β1 integrin was strongly reduced, in agreement with previous studies on the GL-Lect driven internalization of the protein. In contrast, upon prolonged inhibitor treatment, the uptake of α5β1 integrin was increased, and the protein was now internalized by alternative pathways into large perinuclear endosomes. Our findings suggest that under these prolonged inhibitor treatment conditions, α5β1 integrin containing galectin lattices are dissociated, leading to an altered endocytic compartmentalization. [ABSTRACT FROM AUTHOR]

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term a which is absorbent for addition. We study the problem of enumerating all finite common meadows of order n (that is, common meadows with n elements). This problem turns out to be deeply connected with both the number of finite rings of order n and with the number of a certain kind of partition of positive integers. [ABSTRACT FROM AUTHOR]

A continuum theory of pantographic lattices, based on second-grade elasticity, is presented. The proposed model is able to describe the mechanical behavior of a type of material structure made up of multiple layers of pantographic sheets connected with a third family of fibers. Thus, these materials are characterized by an orthogonal pattern of fibers that can bend, stretch and twist. Numerical experiments illustrate the predictive potential of the model when the material is subjected to different types of mechanical loads, including compression, torsion and two kinds of bending. Analyzing the material responses for these various tests makes it possible to reveal unusual deformation patterns characteristic of such "pantographic blocks." Numerical simulations using the finite element method are intended to assist in designing an experimental program using 3D-printed specimens made of different materials. [ABSTRACT FROM AUTHOR]

Let K be a totally real number field of degree n over ℚ , with discriminant and regulator Δ K , R K , respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by | Δ K | − 1 ∕ 2 γ K n exp (2 n ρ ∞ (Λ K)) , where Δ K is the discriminant of the field K, γ K is the additive Hermite–Humbert constant over positive-definite unary forms for K and ρ ∞ (Λ K) is the covering radius of the log-unit lattice. In particular, when K is Galois over ℚ and n is a prime number, the number of homothety classes of unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n) R K 1 n − 1 )) | Δ K | 1 ∕ 2 , where R K is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n))) | Δ K | 1 ∕ 2 . [ABSTRACT FROM AUTHOR]

The existence of a Fourier basis with frequencies in R d for the space of square integrable functions supported on a given parallelepiped in R d , has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in R d to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in R d , a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix. [ABSTRACT FROM AUTHOR]

Polynomial commitments schemes are a powerful tool that enables one party to commit to a polynomial p of degree d, and prove that the committed function evaluates to a certain value z at a specified point u, i.e. p (u) = z , without revealing any additional information about the polynomial. Recently, polynomial commitments have been extensively used as a cryptographic building block to transform polynomial interactive oracle proofs (PIOPs) into efficient succinct arguments. In this paper, we propose a lattice-based polynomial commitment that achieves succinct proof size and verification time in the degree d of the polynomial. Extractability of our scheme holds in the random oracle model under a natural ring version of the BASIS assumption introduced by Wee and Wu (EUROCRYPT 2023). Unlike recent constructions of polynomial commitments by Albrecht et al. (CRYPTO 2022), and by Wee and Wu, we do not require any expensive preprocessing steps, which makes our scheme particularly attractive as an ingredient of a PIOP compiler for succinct arguments. We further instantiate our polynomial commitment, together with the Marlin PIOP (EUROCRYPT 2020), to obtain a publicly-verifiable trusted-setup succinct argument for Rank-1 Constraint System (R1CS). Performance-wise, we achieve 17 MB proof size for 2 20 constraints, which is 15 X smaller than currently the only publicly-verifiable lattice-based SNARK proposed by Albrecht et al. [ABSTRACT FROM AUTHOR]

In recent years, there has been a proliferation of algebraically structured Learning With Errors (LWE) variants, including Ring-LWE, Module-LWE, Polynomial-LWE, Order-LWE, and Middle-Product LWE, and a web of reductions to support their hardness, both among these problems themselves and from related worst-case problems on structured lattices. However, these reductions are often difficult to interpret and use, due to the complexity of their parameters and analysis, and most especially their (frequently large) blowup and distortion of the error distributions. In this paper, we unify and simplify this line of work. First, we give a general framework that encompasses all proposed LWE variants (over commutative base rings) and in particular unifies all prior “algebraic” LWE variants defined over number fields. We then use this framework to give much simpler, more general, and tighter reductions from Ring-LWE to other algebraic LWE variants, including Module-LWE, Order-LWE, and Middle-Product LWE. In particular, all of our reductions have easy-to-analyze and frequently small error expansion; in most cases, they even leave the error unchanged. A main message of our work is that it is straightforward to use the hardness of the original Ring-LWE problem as a foundation for the hardness of all other algebraic LWE problems defined over number fields, via simple and rather tight reductions. [ABSTRACT FROM AUTHOR]

Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k is a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. Our main tool is the Mordell–Weil lattice, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms. [ABSTRACT FROM AUTHOR]

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles. [ABSTRACT FROM AUTHOR]

Let G be a finite p-group with normal subgroup N of order p . The first author and Zalesskii have previously shown that a Z p G -lattice is a permutation module if, and only if, its N -invariants, its N -coinvariants, and a third module are all G/N permutation modules over Z p , Z p and Z p respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a Z p G -lattice for an abelian p -group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the N -invariants and the N -coinvariants of a given lattice U are permutation modules, then so is U . Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice. [ABSTRACT FROM AUTHOR]

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [ABSTRACT FROM AUTHOR]

Hermite–Korkin–Zolotarev (HKZ) reduction is an important notion of lattice reduction which plays a significant role in number theory (particularly the geometry of numbers), and more recently in coding theory and post-quantum cryptography. In this work, we determine a sharp upper bound on the orthogonality defect of HKZ reduced bases up to dimension 3. Using this result, we determine a general upper bound for the orthogonality defect of HKZ reduced bases of arbitrary rank. This upper bound is sharper than existing bounds in literature, such as the one determined by Lagarias, Lenstra and Schnorr [3]. [ABSTRACT FROM AUTHOR]

The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

The swift advancement of quantum computing devices holds the potential to create robust machines that can tackle an extensive array of issues beyond the scope of conventional computers. Consequently, quantum computing machines create new risks at a velocity and scale never seen before, especially with regard to encryption. Lattice-based cryptography is regarded as post-quantum cryptography’s future and a competitor to a quantum computer attack. Thus, there are several advantages to lattice-based cryptographic protocols, including security, effectiveness, reduced energy usage and speed. In this work, we study the learning with errors (LWE) problem and the cryptosystems that are based on the LWE problem and, in addition, we present a new efficient variant of LWE cryptographic scheme.

For any given full rank lattice Λ and natural number K, we consider the point set P (K) = Λ / K ∩ (0 , 1) 2 , with N = # P (K) ≈ K 2 , and bound the spherical cap discrepancy of the projection of these points under the Lambert map to the unit sphere. The bound is of order 1 / N , with leading coefficient given explicitly and depending on Λ only. The proof is established using a lemma that bounds the number of intersections of certain curves with fundamental domains that tile R 2 , and even allows for local perturbations of Λ without affecting the bound, proving to be suitable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date. [ABSTRACT FROM AUTHOR]

For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice. [ABSTRACT FROM AUTHOR]

AbstractTo improve energy absorption and efficiency and develop a structure with outstanding energy absorption capability, this study proposed designing multi-cell tubes filled with triply periodic minimal surface (TPMS) lattices. Based on experimental and simulation methods, the crashworthiness of multi-cell tubes filled with three types of TPMS lattices (Diamond, Gyroid, and Primitive) was investigated. Five multi-cell tube structures were fabricated by increasing the thin-walled tube corner units, improving the energy absorption efficiency by 67.98%. The results show that due to the coupling between TPMS lattices and thin-walled tubes, multi-cell tubes filled with TPMS lattices exhibited better crashworthiness performance. The energy absorption of multi-cell tubes filled with Diamond, Gyroid, and Primitive lattices increased sequentially. The energy absorption per unit mass and energy absorption capability were most notably enhanced for the multi-cell tubes filled with Diamond lattice, with energy absorption efficiency 17.24% higher and energy absorption 104.26% greater than empty multi-cell tubes. This inspires crash designs of rail vehicles and automotive fronts. [ABSTRACT FROM AUTHOR]

In a recent paper by Curran and Goldmakher, the cardinality of h -fold sumsets h A was given when A ⊂ ℤ d has d + 2 elements. In this paper we provide a different method for doing this and obtain a more general result. We also obtain an upper bound for the value of | h A | when A ⊂ ℤ d is a set of d + 3 elements with simplicial hull. [ABSTRACT FROM AUTHOR]

A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparability graphs. This study is motivated by recent results (Corneil et al., SIAM J. Comput. 42(3):792–807, 2013; Dusart et al., SIAM J. Comput. 50(3):1148–1153, 2021; Dusart and Habib, Discret. Appl. Math. 216:149–161, 2017) that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph G is also a maximal chordal subgraph of G. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an algorithm to find in linear time a maximal interval subgraph of a cocomparability graph which improves on the current state of knowledge. Although computing all simplicial vertices is known to be equivalent to the recognition of triangle-free graphs or boolean multiplication, see (Krastch and Spinrad, SIAM J. Comput. 36(2):310–325, 2007; Coudert and Ducoffe, SIAM J. Discrete Math. 32(1):682–694, 2018), we will show that our structural insights in cocomparability graphs together with the definition of a new graph search, allow us to achieve linear time on this class of graphs. [ABSTRACT FROM AUTHOR]

Every stationary action of a strongly irreducible lattice or commensurator of such a lattice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely. Consequently, every minimal action of such a lattice on an infinite compact metric space is topologically free. [ABSTRACT FROM AUTHOR]

This paper explores the application of rough set theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate different forms of exact and rough lattices. Building on Zhou's work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science's prominent lattices and Pawlak's rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations' adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific efficacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational efficiency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved efficiency. The study enhances the understanding of integrating lattices and rough sets, offering potential applications in data analysis, decision support systems, and computational modeling.

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic.

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploscica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics.