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Journal articles on the topic 'Lattice quotients'

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Author: Grafiati

Published: 25 May 2024

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1

Mühle, Henri. "Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group." Annals of Combinatorics 25, no. 2 (April 10, 2021): 307–44. http://dx.doi.org/10.1007/s00026-021-00532-9.

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AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

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Dubsky, Brendan. "Incidence Category of the Young Lattice, Injections Between Finite Sets, and Koszulity." Algebra Colloquium 28, no. 02 (May 11, 2021): 195–212. http://dx.doi.org/10.1142/s1005386721000171.

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We study the quadratic quotients of the incidence category of the Young lattice defined by the zero relations corresponding to adding two boxes to the same row, or to the same column, or both. We show that the last quotient corresponds to the Koszul dual of the original incidence category, while the first two quotients are, in a natural way, Koszul duals of each other and hence they are in particular Koszul self-dual. Both of these two quotients are known to be basic representatives in the Morita equivalence class of the category of injections between finite sets. We also present a new, rather direct, argument establishing this Morita equivalence.

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Thumbakara, Rajesh K. "On Intuitionistic Fuzzy Filters of Intuitionistic Fuzzy Coframes." Journal of Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/793824.

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Frame theory is the study of topology based on its open set lattice, and it was studied extensively by various authors. In this paper, we study quotients of intuitionistic fuzzy filters of an intuitionistic fuzzy coframe. The quotients of intuitionistic fuzzy filters are shown to be filters of the given intuitionistic fuzzy coframe. It is shown that the collection of all intuitionistic fuzzy filters of a coframe and the collection of all intutionistic fuzzy quotient filters of an intuitionistic fuzzy filter are coframes.

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4

Alvarado-García, Alejandro, César Cejudo-Castilla, Hugo Alberto Rincón-Mejía, and Ivan Fernando Vilchis-Montalvo. "Pseudocomplements and strong pseudocomplements in lattices of module classes." Journal of Algebra and Its Applications 17, no. 01 (January 2018): 1850016. http://dx.doi.org/10.1142/s0219498818500160.

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In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices [Formula: see text]-tors (of hereditary torsion classes), [Formula: see text]-nat (the lattice of natural classes) and [Formula: see text]-conat (the lattice of conatural classes) coincide.

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5

Wójtowicz, Marek. "The lattice-isometric copies ofℓ∞(Γ)in quotients of Banach lattices." International Journal of Mathematics and Mathematical Sciences 2003, no. 47 (2003): 3003–6. http://dx.doi.org/10.1155/s0161171203210528.

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LetEbe a Banach lattice and letMbe a norm-closed and Dedekindσ-complete ideal ofE. IfEcontains a lattice-isometric copy ofℓ∞, thenE/Mcontains such a copy as well, orMcontains a lattice copy ofℓ∞. This is one of the consequences of more general results presented in this paper.

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6

Pilaud, Vincent. "Brick polytopes, lattice quotients, and Hopf algebras." Journal of Combinatorial Theory, Series A 155 (April 2018): 418–57. http://dx.doi.org/10.1016/j.jcta.2017.11.014.

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7

Ma, Jingjing, and R. H. Redfield. "Fields of quotients of lattice-ordered domains." algebra universalis 52, no. 4 (February 2005): 383–401. http://dx.doi.org/10.1007/s00012-004-1875-z.

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8

Demonet, Laurent, Osamu Iyama, Nathan Reading, Idun Reiten, and Hugh Thomas. "Lattice theory of torsion classes: Beyond 𝜏-tilting theory." Transactions of the American Mathematical Society, Series B 10, no. 18 (April 25, 2023): 542–612. http://dx.doi.org/10.1090/btran/100.

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The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set t o r s A \mathsf {tors} A of torsion classes over a finite-dimensional algebra A A . We show that t o r s A \mathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of t o r s A \mathsf {tors} A . In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that t o r s A \mathsf {tors} A is completely congruence uniform. When I I is a two-sided ideal of A A , t o r s ( A / I ) \mathsf {tors} (A/I) is a lattice quotient of t o r s A \mathsf {tors} A which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of t o r s A \mathsf {tors} A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras Π \Pi , for which t o r s Π \mathsf {tors} \Pi is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between t o r s k Q \mathsf {tors} k Q and the Cambrian lattice when Q Q is a Dynkin quiver. We also prove that, in type A A , the algebraic quotients of t o r s Π \mathsf {tors} \Pi are exactly its Hasse-regular lattice quotients.

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9

KAKARIADIS, EVGENIOS T. A. "Finite-dimensional approximations for Nica–Pimsner algebras." Ergodic Theory and Dynamical Systems 40, no. 12 (August 9, 2019): 3375–402. http://dx.doi.org/10.1017/etds.2019.44.

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We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

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10

Jenča, G., and S. Pulmannová. "Ideals and quotients in lattice ordered effect algebras." Soft Computing 5, no. 5 (October 2001): 376–80. http://dx.doi.org/10.1007/s005000100139.

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11

Dermenjian, Aram, Christophe Hohlweg, and Vincent Pilaud. "The facial weak order and its lattice quotients." Transactions of the American Mathematical Society 370, no. 2 (October 24, 2017): 1469–507. http://dx.doi.org/10.1090/tran/7307.

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12

MEINEL, JOANNA, and CATHARINA STROPPEL. "GOLDIE RANK OF PRIMITIVE QUOTIENTS VIA LATTICE POINT ENUMERATION." Glasgow Mathematical Journal 55, A (October 2013): 149–68. http://dx.doi.org/10.1017/s0017089513000566.

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AbstractLet k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

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13

Wilson, S. M. J. "The isomorphism class of a set of lattices." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 2 (March 1989): 197–210. http://dx.doi.org/10.1017/s0305004100067682.

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AbstractLet R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

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14

Darnel, Michael R. "Value quotients and product torsion classes of lattice-ordered groups." Algebra universalis 60, no. 4 (April 14, 2009): 405–23. http://dx.doi.org/10.1007/s00012-009-2111-7.

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15

Deitmar, Anton, and Werner Hoffman. "Spectral Estimates for Towers of Noncompact Quotients." Canadian Journal of Mathematics 51, no. 2 (April 1, 1999): 266–93. http://dx.doi.org/10.4153/cjm-1999-014-3.

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AbstractWe prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ.

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16

Liao, Benben. "Strong Banach property (T) for simple algebraic groups of higher rank." Journal of Topology and Analysis 06, no. 01 (February 13, 2014): 75–105. http://dx.doi.org/10.1142/s1793525314500010.

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We extend Vincent Lafforgue's results to Sp4. As applications, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1, and any affine isometric action of such a group, or of any cocompact lattice in it, in a Banach space of type > 1 has a fixed point.

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17

Düzlü, Samed, and Juliane Krämer. "Application of automorphic forms to lattice problems." Journal of Mathematical Cryptology 16, no. 1 (January 1, 2022): 156–97. http://dx.doi.org/10.1515/jmc-2021-0045.

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Abstract In this article, we propose a new approach to the study of lattice problems used in cryptography. We specifically focus on module lattices of a fixed rank over some number field. An essential question is the hardness of certain computational problems on such module lattices, as the additional structure may allow exploitation. The fundamental insight is the fact that the collection of those lattices are quotients of algebraic manifolds by arithmetic subgroups. Functions in these spaces are studied in mathematics as part of the number theory. In particular, those form a module over the Hecke algebra associated with the general linear group. We use results on these function spaces to define a class of distributions on the space of lattices. Using the Hecke algebra, we define Hecke operators associated with collections of prime ideals of the number field and show a criterion on distributions to converge to the uniform distribution, if the Hecke operators are applied to the chosen distribution. Our approach is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski (CRYPTO’20) on self-reduction of ideal lattices via Arakelov divisors.

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18

Giraudet, M., and J. K. Truss. "Recovering ordered structures from quotients of their automorphism groups." Journal of Symbolic Logic 68, no. 4 (December 2003): 1189–98. http://dx.doi.org/10.2178/jsl/1067620181.

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AbstractWe show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

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MARGULIS, GREGORY A., and NANTIAN QIAN. "Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices." Ergodic Theory and Dynamical Systems 21, no. 1 (February 2001): 121–64. http://dx.doi.org/10.1017/s0143385701001109.

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Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

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20

Momot, Aleksander. "On Modular Ball-Quotient Surfaces of Kodaira Dimension One." ISRN Geometry 2011 (June 19, 2011): 1–5. http://dx.doi.org/10.5402/2011/214853.

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Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].

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21

Perera, Francesc. "Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero." Canadian Journal of Mathematics 53, no. 3 (June 1, 2001): 592–630. http://dx.doi.org/10.4153/cjm-2001-025-2.

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AbstractWe give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of themultiplier algebra , is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of is reflected in the fact that can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

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22

Conder, Marston. "Two new proofs of the fact that triangle groups are distinguished by their finite quotients." New Zealand Journal of Mathematics 52 (March 3, 2022): 827–44. http://dx.doi.org/10.53733/193.

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In a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $\Gamma$ is a finitely-generated Fuchsian group and $\Sigma$ is a lattice in a connected Lie group, such that $\Gamma$ and $\Sigma$ have exactly the same finite quotients, then $\Gamma$ is isomorphic to $\Sigma$. As a consequence, two triangle groups $\Delta(r,s,t)$ and $\Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').

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Kasparian, Azniv. "Projective Embeddings of Ball Quotients, Birational to a Bi-elliptic Surface." Proceedings of the Bulgarian Academy of Sciences 76, no. 1 (January 30, 2023): 3–11. http://dx.doi.org/10.7546/crabs.2023.01.01.

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For a neat lattice $$\Gamma < SU(1,2)$$, whose quotient $${\mathbb B} / \Gamma$$ is birational to a bi-elliptic surface, we compute the dimensions of the cuspidal $$\Gamma$$-modular forms $$[ \Gamma,n]_o$$ and all modular forms $$[ \Gamma, n]$$ of weight $$n \geq 2. $$ The work provides a sufficient condition for a subspace $$V \subset [ \Gamma, n]$$ to determine a regular projective embedding of the Baily-Borel compactification $$\widehat{ {\mathbb B} / \Gamma}$$ and applies this criterion to a specific example.

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BUENGER, C. DAVIS, and CHENG ZHENG. "Non-divergence of unipotent flows on quotients of rank-one semisimple groups." Ergodic Theory and Dynamical Systems 37, no. 1 (December 28, 2015): 103–28. http://dx.doi.org/10.1017/etds.2015.43.

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Let$G$be a semisimple Lie group of rank one and$\unicode[STIX]{x1D6E4}$be a torsion-free discrete subgroup of$G$. We show that in$G/\unicode[STIX]{x1D6E4}$, given$\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than$\unicode[STIX]{x1D6FF}$for a$1-\unicode[STIX]{x1D716}$proportion of the time, for some$\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup$\unicode[STIX]{x1D6E4}$and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces.Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed$\unicode[STIX]{x1D716}>0$, there exists an injectivity radius$\unicode[STIX]{x1D6FF}$such that, for any unipotent trajectory$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a$1-\unicode[STIX]{x1D716}$proportion of the time in the set with injectivity radius larger than$\unicode[STIX]{x1D6FF}$, for all large$T>0$, or there exists a$\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup$L$of$G$which intersects$g\unicode[STIX]{x1D6E4}g^{-1}$in a small covolume lattice. We also extend these results to when$G$is the product of rank-one semisimple groups and$\unicode[STIX]{x1D6E4}$a discrete subgroup of$G$whose projection onto each non-trivial factor is torsion free.

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Fabbri, Marc A., and Frank Okoh. "Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras." Canadian Journal of Mathematics 51, no. 3 (June 1, 1999): 523–45. http://dx.doi.org/10.4153/cjm-1999-024-x.

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AbstractVirasoro-toroidal algebras, , are semi-direct products of toroidal algebras and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let Γ be an extension of a simply laced lattice by a hyperbolic lattice of rank two. There is a Fock space V(Γ) corresponding to Γ with a decomposition as a complex vector space: V(Γ) = . Fabbri and Moody have shown that when m ≠ 0, K(m) is an irreducible representation of . In this paper we produce a filtration of -submodules of K(0). When L is an arbitrary geometric lattice and n is a positive integer, we construct a Virasoro-Heisenberg algebra . Let Q be an extension of by a degenerate rank one lattice. We determine the components of V(Γ) that are irreducible -modules and we show that the reducible components have a filtration of -submodules with completely reducible quotients. Analogous results are obtained for . These results complement and extend results of Fabbri and Moody.

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Garbagnati, Alice. "On K3 Surface Quotients of K3 or Abelian Surfaces." Canadian Journal of Mathematics 69, no. 02 (April 2017): 338–72. http://dx.doi.org/10.4153/cjm-2015-058-1.

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Abstract The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases. Moreover, we prove that a K3 surface XG is the minimal model of the quotient of an Abelian surface by a group G if and only if a certain configuration of rational curves is present on XG . Again, this result was known only in some special cases, in particular, if G has order 2 or 3.

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Matsumoto, Kengo. "C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems." Journal of the Australian Mathematical Society 81, no. 3 (December 2006): 369–85. http://dx.doi.org/10.1017/s1446788700014373.

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AbstractA λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. InDoc. Math.7 (2002) 1–30, the author constructed aC*-algebraO£associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebraO£, under a certain condition on £ called (II). As a result, the class of theC*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

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Beshkov, Pancho, Azniv Kasparian, and Gregory Sankaran. "Saturated and primitive smooth compactifications of ball quotients." Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics 106 (December 12, 2019): 53–77. http://dx.doi.org/10.60063/gsu.fmi.106.53-77.

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Let $X = ( {\mathbb B} / \Gamma)'$ be a smooth toroidal compactification of a quotient of the complex $2$-ball ${\mathbb B} = {\rm PSU} _{2,1} / {\rm PS} (U_2 \times U_1)$ by a lattice $\Gamma < {\rm PSU} _{2,1}$, $D := X \setminus ( \mathbb{B} /\Gamma)$ be the toroidal compactifying divisor of $X$, $\rho : X \rightarrow Y$ be a finite composition of blow downs to a minimal surface $Y$ and $E(\rho)$ be the exceptional divisor of $\rho$. The present article establishes a bijective correspondence between the finite unramified coverings of ordered triples $(X, D, E)$ and the finite unramified coverings of $( \rho (X), \rho (D), \rho (E)).$ We say that $(X, D,E(\rho))$ is saturated if all the unramified coverings $f: (X', D', E' (\rho')) \rightarrow (X, D, E)$ are isomorphisms, while $(X, D, E (\rho))$ is primitive exactly when any unramified covering $f: (X, D, E (\rho)) \rightarrow ( f(X), f(D), f(E(\rho)))$ is an isomorphism. The covering relations among the smooth toroidal compactifications $(\mathbb{B} / \Gamma)'$ are studied by Uludag's [7], Stover's [6], Di Cerbo and Stover's [2] and other articles.In the case of a single blow up $\rho = \beta : X = ( {\mathbb B} / \Gamma )' \rightarrow Y$ of finitely many points of $Y$, we show that there is an isomorphism $\Phi : {\rm Aut} (Y, \beta (D)) \rightarrow {\rm Aut} (X, D)$ of the relative automorphism groups and ${\rm Aut} (X, D)$ is a finite group. Moreover, when $Y$ is an abelian surface then any finite unramified covering $f: (X, D, E( \beta)) \rightarrow ( f(X), f(D), f (E( \beta)))$ factors through an ${\rm Aut} (X, D)$-Galois covering. We discuss the saturation and the primitiveness of $X$ with Kodaira dimension $\kappa (X)= - \infty$, as well as of $X$ with $K3$ or Enriques minimal model $Y$.

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Mok, Ngaiming. "Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball." Compositio Mathematica 155, no. 11 (September 19, 2019): 2129–49. http://dx.doi.org/10.1112/s0010437x19007577.

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We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

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DROSTE, MANFRED, and DIETRICH KUSKE. "Recognizable languages in divisibility monoids." Mathematical Structures in Computer Science 11, no. 6 (December 2001): 743–70. http://dx.doi.org/10.1017/s0960129501003395.

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We define the class of divisibility monoids that arise as quotients of the free monoid Σ* modulo certain equations of the form ab = cd. These form a much larger class than free partially commutative monoids, and we show, under certain assumptions, that the recognizable languages in these divisibility monoids coincide with c-rational languages. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to these monoids. We obtain Ochmański's theorem on recognizable languages in free partially commutative monoids as a consequence.

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GUÀRDIA, J. "EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3." Journal of the London Mathematical Society 64, no. 2 (October 2001): 299–310. http://dx.doi.org/10.1112/s0024610701002538.

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An explicit geometrical study of the curves[formula here]is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.

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32

Ayyer, Arvind, Anne Schilling, Benjamin Steinberg, and Nicolas M. Thiéry. "Markov chains, ${\mathscr R}$-trivial monoids and representation theory." International Journal of Algebra and Computation 25, no. 01n02 (February 2015): 169–231. http://dx.doi.org/10.1142/s0218196715400081.

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We develop a general theory of Markov chains realizable as random walks on [Formula: see text]-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom–Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

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33

YEUNG, SAI-KEE. "INTEGRALITY AND ARITHMETICITY OF CO-COMPACT LATTICE CORRESPONDING TO CERTAIN COMPLEX TWO-BALL QUOTIENTS OF PICARD NUMBER ONE." Asian Journal of Mathematics 8, no. 1 (2004): 107–30. http://dx.doi.org/10.4310/ajm.2004.v8.n1.a9.

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34

KAUFMANN, RALPH M. "PATH SPACE DECOMPOSITIONS FOR THE VIRASORO ALGEBRA AND ITS VERMA MODULES." International Journal of Modern Physics A 10, no. 07 (March 20, 1995): 943–61. http://dx.doi.org/10.1142/s0217751x95000462.

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Starting from a detailed analysis of the structure of path spaces of the [Formula: see text] fusion graphs and the corresponding irreducible Virasoro algebra quotients V(c, h) for the (2, q odd) models, we introduce the notion of an admissible path space representation. The path spaces [Formula: see text] over the [Formula: see text] graphs are isomorphic to the path spaces over Coxeter A graphs that appear in FB models. We give explicit construction algorithms for admissible representations. From the finite-dimensional results of these algorithms we derive a decomposition of V(c, h) into its positive and negative definite subspaces w.r.t. the Shapovalov form and the corresponding signature characters. Finally, we treat the Virasoro operation on the lattice induced by admissible representations, adopting a particle point of view. We use this analysis to decompose the Virasoro algebra generators themselves. This decomposition also takes into account the nonunitarity of the (2, q) models.

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35

HEERSINK, BYRON. "Equidistribution of Farey sequences on horospheres in covers of and applications." Ergodic Theory and Dynamical Systems 41, no. 2 (October 7, 2019): 471–93. http://dx.doi.org/10.1017/etds.2019.71.

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We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

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36

Dan, Christina-Theresia. "A Connection Between the Reticulation of a Ring of Quotients and the Localization Lattice of the Reticulation of a Commutative Ring." Communications in Algebra 35, no. 6 (May 29, 2007): 1783–807. http://dx.doi.org/10.1080/00927870701246502.

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37

Sorbi, Andrea. "Some Quotient Lattices of the Medvedev Lattice." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 37, no. 9-12 (1991): 167–82. http://dx.doi.org/10.1002/malq.19910370905.

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38

MALONEY, GREGORY R., and DAN RUST. "Beyond primitivity for one-dimensional substitution subshifts and tiling spaces." Ergodic Theory and Dynamical Systems 38, no. 3 (September 20, 2016): 1086–117. http://dx.doi.org/10.1017/etds.2016.58.

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We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

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39

Bootland, Carl, Wouter Castryck, Alan Szepieniec, and Frederik Vercauteren. "A framework for cryptographic problems from linear algebra." Journal of Mathematical Cryptology 14, no. 1 (July 21, 2020): 202–17. http://dx.doi.org/10.1515/jmc-2019-0032.

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AbstractWe introduce a general framework encompassing the main hard problems emerging in lattice-based cryptography, which naturally includes the recently proposed Mersenne prime cryptosystem, but also problems coming from code-based cryptography. The framework allows to easily instantiate new hard problems and to automatically construct plausibly post-quantum secure primitives from them. As a first basic application, we introduce two new hard problems and the corresponding encryption schemes. Concretely, we study generalisations of hard problems such as SIS, LWE and NTRU to free modules over quotients of ℤ[X] by ideals of the form (f, g), where f is a monic polynomial and g ∈ ℤ[X] is a ciphertext modulus coprime to f. For trivial modules (i.e. of rank one), the case f = Xn + 1 and g = q ∈ ℤ>1 corresponds to ring-LWE, ring-SIS and NTRU, while the choices f = Xn – 1 and g = X – 2 essentially cover the recently proposed Mersenne prime cryptosystems. At the other extreme, when considering modules of large rank and letting deg(f) = 1, one recovers the framework of LWE and SIS.

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40

de Pagter, Ben, and Anthony W. Wickstead. "Free and projective Banach lattices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 1 (January 30, 2015): 105–43. http://dx.doi.org/10.1017/s0308210512001709.

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We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

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41

Futa, Yuichi, Hiroyuki Okazaki, and Yasunari Shidama. "Quotient Module of Z-module." Formalized Mathematics 20, no. 3 (December 1, 2012): 205–14. http://dx.doi.org/10.2478/v10037-012-0024-y.

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Summary In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module.

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42

Saeid, Arsham Borumand, and Roghayeh Koohnavard. "On residuated skew lattices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 1 (March 1, 2019): 245–68. http://dx.doi.org/10.2478/auom-2019-0013.

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Abstract In this paper, we define residuated skew lattice as non-commutative generalization of residuated lattice and investigate its properties. We show that Green’s relation 𝔻 is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. Deductive system and skew deductive system in residuated skew lattices are defined and relationships between them are given and proved. We define branchwise residuated skew lattice and show that a conormal distributive residuated skew lattice is equivalent with a branchwise residuated skew lattice under a condition.

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43

Holdon, Liviu-Constantin, and Arsham Borumand Saeid. "Ideals of Residuated Lattices." Studia Scientiarum Mathematicarum Hungarica 58, no. 2 (June 29, 2021): 182–205. http://dx.doi.org/10.1556/012.2021.58.2.1493.

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In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

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44

Ball, Richard N., and Anthony W. Hager. "Epicompletetion of archimedean l–groups and vector lattices with weak unit." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 48, no. 1 (February 1990): 25–56. http://dx.doi.org/10.1017/s1446788700035175.

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AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

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Stewart, Ian. "Exotic Patterns of Synchrony in Planar Lattice Networks." International Journal of Bifurcation and Chaos 29, no. 02 (February 2019): 1930003. http://dx.doi.org/10.1142/s0218127419300039.

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Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.

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46

Piciu, Dana, Christina Theresia Dan, and Florentina Chirteş. "Some decompositions of filters in residuated lattices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 1 (March 1, 2019): 211–31. http://dx.doi.org/10.2478/auom-2019-0011.

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Abstract In this paper we introduce a new class of residuated lattice: residuated lattice with (C∧&→) property and we prove that (C∧&→) ⇔ (C→) + (C∧).Also, we introduce and characterize C→, C∨, C∧ and C∧ & → filters in residuated lattices (i.e., we characterize the filters for which the quotient algebra that is constructed via these filters is a residuated lattice with C→ (C∨ or C∧ or C∧&→ property). We state and prove some results which establish the relationships between these filters and other filters of residuated lattices: BL filters, MTL filters, divisible filters and, by some examples, we show that these filters are different. Starting from the results of algebras, we present for MTL filters, BL filters and C∧&→ filters the decomposition conditions.

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47

ALBU, TOMA, MIHAI IOSIF, and MARK L. TEPLY. "MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES." Journal of Algebra and Its Applications 03, no. 04 (December 2004): 391–410. http://dx.doi.org/10.1142/s0219498804000939.

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A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

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48

ZHAO, TIEHONG. "A minimal volume arithmetic cusped complex hyperbolic orbifold." Mathematical Proceedings of the Cambridge Philosophical Society 150, no. 2 (October 8, 2010): 313–42. http://dx.doi.org/10.1017/s0305004110000526.

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AbstractThe sister of Eisenstein–Picard modular group is described explicitly in [10], whose quotient is a noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume (see [16]). We give a construction of a fundamental domain for this group. A presentation of that lattice can be obtained from that construction, which relates to one of Mostow's lattices.

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49

Johnson, E. W., Johnny A. Johnson, and Monty B. Taylor. "p-systems in local Noether lattices." International Journal of Mathematics and Mathematical Sciences 17, no. 4 (1994): 655–60. http://dx.doi.org/10.1155/s016117129400092x.

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In this paper we introduce the concept of ap-system in a local Noether lattice and obtain several characterizations of these elements. We first obtain a topological characterization and then a characterization in terms of the existence of a certain type of decreasing sequence of elements. In addition,p-systems are characterized in quotient lattices and completions.

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50

Piciu, Dana, Christina Theresia Dan, and Anca Dina. "Gődel filters in residuated lattices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 1 (March 1, 2021): 183–200. http://dx.doi.org/10.2478/auom-2021-0012.

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Abstract In this paper, in the spirit of [4], we study a new type of filters in residuated lattices : Gődel filters. So, we characterize the filters for which the quotient algebra that is constructed via these filters is a Gődel algebra and we establish the connections between these filters and other types of filters. Using Gődel filters we characterize the residuated lattices which are Gődel algebras. Also, we prove that a residuated lattice is a Gődel algebra (divisible residuated lattice, MTL algebra, BL algebra) if and only if every filter is a Gődel filter (divisible filter, MTL filter, BL filter). Finally, we present some results about injective Gődel algebras showing that complete Boolean algebras are injective objects in the category of Gődel algebras.

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