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Journal articles on the topic 'Lattices theory'

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Published: 25 May 2024

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1

Day, Alan. "Doubling Constructions in Lattice Theory." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 252–69. http://dx.doi.org/10.4153/cjm-1992-017-7.

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AbstractThis paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

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2

Flaut, Cristina, Dana Piciu, and Bianca Liana Bercea. "Some Applications of Fuzzy Sets in Residuated Lattices." Axioms 13, no. 4 (April 18, 2024): 267. http://dx.doi.org/10.3390/axioms13040267.

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Many papers have been devoted to applying fuzzy sets to algebraic structures. In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory, lattice theory, and coding theory points of view, and some applications of fuzzy sets in residuated lattices are presented. Since ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic, first, we investigate the lattice of fuzzy ideals in residuated lattices and study some connections between fuzzy sets associated to ideals and Hadamard codes. Finally, we present applications of fuzzy sets in coding theory.

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3

Harremoës, Peter. "Entropy Inequalities for Lattices." Entropy 20, no. 10 (October 12, 2018): 784. http://dx.doi.org/10.3390/e20100784.

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We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

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4

Ježek, J., P. PudláK, and J. Tůma. "On equational theories of semilattices with operators." Bulletin of the Australian Mathematical Society 42, no. 1 (August 1990): 57–70. http://dx.doi.org/10.1017/s0004972700028148.

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In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

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5

Frapolli, Nicolò, Shyam Chikatamarla, and Ilya Karlin. "Theory, Analysis, and Applications of the Entropic Lattice Boltzmann Model for Compressible Flows." Entropy 22, no. 3 (March 24, 2020): 370. http://dx.doi.org/10.3390/e22030370.

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The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both these problems are presented by modifying the original lattices without increasing the number of discrete velocities and without altering the numerical algorithm. In order to increase the Mach number, we employ shifted lattices while the magnitude of lattice speeds is increased in order to extend the temperature range. Accuracy and efficiency of the shifted lattices are demonstrated with simulations of the supersonic flow field around a diamond-shaped and NACA0012 airfoil, the subsonic, transonic, and supersonic flow field around the Busemann biplane, and the interaction of vortices with a planar shock wave. For the lattices with extended temperature range, the model is validated with the simulation of the Richtmyer–Meshkov instability. We also discuss some key ideas of how to reduce the number of discrete speeds in three-dimensional simulations by pruning of the higher-order lattices, and introduce a new construction of the corresponding guided equilibrium by entropy minimization.

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6

McCulloch, Ryan. "Finite groups with a trivial Chermak–Delgado subgroup." Journal of Group Theory 21, no. 3 (May 1, 2018): 449–61. http://dx.doi.org/10.1515/jgth-2017-0042.

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Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the Chermak–Delgado subgroup of G. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p-group lattices play a major role in the author’s constructions.

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7

Grabowski, Adam. "Stone Lattices." Formalized Mathematics 23, no. 4 (December 1, 2015): 387–96. http://dx.doi.org/10.1515/forma-2015-0031.

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Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

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8

Bronzan, J. B. "Hamiltonian lattice gauge theory: wavefunctions on large lattices." Nuclear Physics B - Proceedings Supplements 30 (March 1993): 916–19. http://dx.doi.org/10.1016/0920-5632(93)90356-b.

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9

Ge, Mo-Lin, Liangzhong Hu, and Yiwen Wang. "KNOT THEORY, PARTITION FUNCTION AND FRACTALS." Journal of Knot Theory and Its Ramifications 05, no. 01 (February 1996): 37–54. http://dx.doi.org/10.1142/s0218216596000047.

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In this paper we first provide the open chain and the closed chain method to calculate the partition functions of the typical fractal lattices, i.e. a special kind of Sierpinski carpets(SC) and the triangular Sierpinski gaskets(SG). We then apply knot theory to fractal lattices by changing lattice graphs into link diagrams according to the interaction models, and explicitly obtain the partition functions of a special SC for the edge interaction models. These partition functions are also the knot invariants of the corresponding link diagrams. This is the first time that topology enters into fractals.

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10

NEBE, GABRIELE. "ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES." International Journal of Number Theory 09, no. 08 (December 2013): 1933–59. http://dx.doi.org/10.1142/s179304211350067x.

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The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

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11

Futa, Yuichi, and Yasunari Shidama. "Lattice of ℤ-module." Formalized Mathematics 24, no. 1 (March 1, 2016): 49–68. http://dx.doi.org/10.1515/forma-2016-0005.

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Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

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12

Ballal, Sachin, and Vilas Kharat. "Zariski topology on lattice modules." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550066. http://dx.doi.org/10.1142/s1793557115500667.

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Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

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13

Borcherds, Richard E. "Lattices like the Leech lattice." Journal of Algebra 130, no. 1 (April 1990): 219–34. http://dx.doi.org/10.1016/0021-8693(90)90110-a.

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14

Horváth, Eszter K., Sándor Radeleczki, Branimir Šešelja, and Andreja Tepavčević. "A Note on Cuts of Lattice-Valued Functions and Concept Lattices." Mathematica Slovaca 73, no. 3 (June 1, 2023): 583–94. http://dx.doi.org/10.1515/ms-2023-0043.

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ABSTRACT Motivated by applications of lattice-valued functions (lattice-valued fuzzy sets) in the theory of ordered structures, we investigate a special kind of posets and lattices induced by these mappings. As a framework, we use the Formal Concept Analysis in which these ordered structures can be naturally observed. We characterize the lattice of cut sets and the Dedekind-MacNeille completion of the set of images of a lattice valued function by suitable concept lattices and we give necessary and sufficient conditions under which these lattices coincide. In addition, we give conditions under which the lattice of cuts is completely distributive.

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15

FUKSHANSKY, LENNY, and KATHLEEN PETERSEN. "ON WELL-ROUNDED IDEAL LATTICES." International Journal of Number Theory 08, no. 01 (February 2012): 189–206. http://dx.doi.org/10.1142/s179304211250011x.

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We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

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16

Luo, Congwen. "S-Lattice Congruences of S-Lattices." Algebra Colloquium 19, no. 03 (July 5, 2012): 465–72. http://dx.doi.org/10.1142/s1005386712000326.

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In this paper, the S-lattices are introduced as a representation of lattice-ordered monoids. The smallest S-lattice congruence induced by a relation on an S-lattice is characterized and the correspondence between the S-lattice congruences and S-ideals in an S-distributive lattice is discussed. These generalize some recent results of lattices and lattice-ordered semigroups.

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17

Ježek, Jaroslav, and George F. McNulty. "The existence of finitely based lower covers for finitely based equational theories." Journal of Symbolic Logic 60, no. 4 (December 1995): 1242–50. http://dx.doi.org/10.2307/2275885.

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By an equational theory we mean a set of equations from some fixed language which is closed with respect to logical consequences. We regard equations as universal sentences whose quantifier-free parts are equations between terms. In our notation, we suppress the universal quantifiers. Once a language has been fixed, the collection of all equational theories for that language is a lattice ordered by set inclusion The meet in this lattice is simply intersection; the join of a collection of equational theories is the equational theory axiomatized by the union of the collection. In this paper we prove, for languages with only finitely many fundamental operation symbols, that any nontrivial finitely axiomatizable equational theory covers some other finitely axiomatizable equational theory. In fact, our result is a little more general.There is an extensive literature concerning lattices of equational theories. These lattices are always algebraic. Compact elements of these lattices are the finitely axiomatizable equational theories. We also call them finitely based. The largest element in the lattice is compact; it is the equational theory based on the single equation x ≈ y. The smallest element of the lattice is the trivial theory consisting of tautological equations. For all but the simplest languages, the lattice of equational theories is intricate. R. McKenzie in [6] was able to prove in essence that the underlying language can be recovered from the isomorphism type of this lattice.

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18

Han, Bao Chuan, Ya Jun Du, Chang Wang, and Jing Xu. "A Concept Lattice Merger Approach for Ontology Construction." Advanced Materials Research 181-182 (January 2011): 667–72. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.667.

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The method of merging concept lattice in domain ontology construction can describe the implicit concepts and relationships between concepts more appropriately for semantic representation and query match. In order to enrich semantic query, the paper intends to apply the theory of Formal Concept Analysis (FCA) to establish source concept lattices, through which the domain concepts are extracted from source concept lattices to generate the optimized concept lattice. Then, the ontology tree is generated by lattice mapping ontology algorithm (LMOA) combing some hierarchical relations in the optimized concept lattice. The experiment proves that the domain ontology can be achieved effectively by merging concept lattices and provide the semantic relations more precisely.

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19

Han, Bao Chuan, Ya Jun Du, Chang Wang, and Jing Xu. "A Concept Lattice Merger Approach for Ontology Construction." Advanced Materials Research 181-182 (January 2011): 754–59. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.754.

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The method of merging concept lattice in domain ontology construction can describe the implicit concepts and relationships between concepts more appropriately for semantic representation and query match. In order to enrich semantic query, the paper intends to apply the theory of Formal Concept Analysis (FCA) to establish source concept lattices, through which the domain concepts are extracted from source concept lattices to generate the optimized concept lattice. Then, the ontology tree is generated by lattice mapping ontology algorithm (LMOA) combing some hierarchical relations in the optimized concept lattice. The experiment proves that the domain ontology can be achieved effectively by merging concept lattices and provide the semantic relations more precisely.

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20

Decker, Karsten M., and Philippe de Forcrand. "Pure SU(2) lattice gauge theory on 324 lattices." Nuclear Physics B - Proceedings Supplements 17 (September 1990): 567–70. http://dx.doi.org/10.1016/0920-5632(90)90315-l.

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21

Sinclair, R. "Calculations on infinite lattices applied to lattice gauge theory." Physical Review D 42, no. 12 (December 15, 1990): 4182–85. http://dx.doi.org/10.1103/physrevd.42.4182.

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22

Kubovy, Michael, and Johan Wagemans. "Grouping by Proximity and Multistability in Dot Lattices: A Quantitative Gestalt Theory." Psychological Science 6, no. 4 (July 1995): 225–34. http://dx.doi.org/10.1111/j.1467-9280.1995.tb00597.x.

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Gestalt phenomena have long resisted quantification In the spirit of Gestalt field theory, we propose a theory that predicts the probability of grouping by proximity in the six kinds of dot lattices (hexagonal, rhombic, square, rectangular, centered rectangular, and oblique) We claim that the unstable perceptual organization of dot lattices is caused by competing forces that attract each dot to other dots in its neighborhood We model the decline of these forces as a function of distance with an exponential decay function This attraction function has one parameter, the attraction constant Simple assumptions allow us to predict the entropy of the perceptual organization of different dot lattices We showed dot lattices tachistoscopically to 7 subjects, and from the probabilities of the perceived organizations, we calculated the entropy of each lattice for each subject The model fit the data exceedingly well The attraction constant did not vary much over subjects

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23

KENDZIORRA, ANDREAS, and STEFAN E. SCHMIDT. "NETWORK CODING WITH MODULAR LATTICES." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1319–42. http://dx.doi.org/10.1142/s0219498811005208.

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Kötter and Kschischang presented in 2008 a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map d : (U, V) ↦ dim (U+V)- dim (U∩V). In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map d : (u, v) ↦ h(u ∨ v) - h(u ∧ v), where h is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a Singleton bound for codes, which are subsets of modular lattices.

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24

Dziobiak, Wies?aw. "On lattice identities satisfied in subquasivariety lattices of varieties of modular lattices." Algebra Universalis 22, no. 2-3 (June 1986): 205–14. http://dx.doi.org/10.1007/bf01224026.

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25

de la Maza, Ana Cecilia, and Remo Moresi. "Hermitean (semi) lattices and Rolf’s lattice." Algebra universalis 66, no. 1-2 (August 30, 2011): 49–62. http://dx.doi.org/10.1007/s00012-011-0141-4.

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26

BRAZHNYI, V. A., and V. V. KONOTOP. "THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES." Modern Physics Letters B 18, no. 14 (June 10, 2004): 627–51. http://dx.doi.org/10.1142/s0217984904007190.

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We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

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27

Symonds, Peter. "Relative characters for H-projective RG-lattices." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 2 (September 1988): 207–13. http://dx.doi.org/10.1017/s0305004100065397.

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If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

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28

Pardo-Guerra, Sebastián, Hugo Alberto Rincón-Mejía, and Manuel Gerardo Zorrilla-Noriega. "Some isomorphic big lattices and some properties of lattice preradicals." Journal of Algebra and Its Applications 19, no. 07 (July 24, 2019): 2050140. http://dx.doi.org/10.1142/s0219498820501406.

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According to Albu and Iosif, [2, Definition 1.1] a lattice preradical is a subfunctor of the identity functor on the category [Formula: see text] of linear modular lattices, whose objects are the complete modular lattices and whose morphisms are linear morphisms. In this paper, we describe some big lattices which are isomorphic to the big lattice of lattice preradicals and we study the four classical operations that occur in the lattice of preradicals of modules over a ring [Formula: see text], namely, the join, the meet, the product and the coproduct. We show that some results about the lattice of module preradicals can be extended to the lattice of lattice preradicals. In particular, we show the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical [Formula: see text], as well as some of their properties.

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29

Tewary, V. K., and Robb Thomson. "Lattice statics of interfaces and interfacial cracks in bimaterial solids." Journal of Materials Research 7, no. 4 (April 1992): 1018–28. http://dx.doi.org/10.1557/jmr.1992.1018.

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A method for calculating lattice statics Green's function is described for a bimaterial lattice or a bicrystal containing a plane interface. The method involves creation of two half space lattices containing free surfaces and then joining them to form a bicrystal. The two half space lattices may have different structures as in a two-phase bicrystal or may be of the same type but joined at different orientations to form a grain boundary interface. The method is quite general but, in this paper, has been applied only to a simple model bicrystal formed by two simple cubic lattices with nearest neighbor interactions. The bimaterial Green's function is modified to account for an interfacial crack that is used to calculate the displacement field due to an applied external force. It is found that the displacement field, as calculated by using the lattice theory, does not have the unphysical oscillations predicted by the continuum theory.

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30

SUAREZ, IVAN. "MODULAR LATTICES OVER CM FIELDS." International Journal of Number Theory 05, no. 05 (August 2009): 859–69. http://dx.doi.org/10.1142/s1793042109002420.

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We study some properties of Arakelov-modular lattices, which are particular modular ideal lattices over CM fields. There are two main results in this paper. The first one is the determination of the number of Arakelov-modular lattices of fixed level over a given CM field provided that an Arakelov-modular lattice is already known. This number depends on the class numbers of the CM field and its maximal totally real subfield. The first part gives also a way to compute all these Arakelov-modular lattices. In the second part, we describe the levels that can occur for some multiquadratic CM number fields.

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31

Kusraev, Anatoly, and Semën Kutateladze. "Geometric Characterization of Injective Banach Lattices." Mathematics 9, no. 3 (January 27, 2021): 250. http://dx.doi.org/10.3390/math9030250.

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This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

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32

MANN, CASEY, JENNIFER MCLOUD-MANN, RAMONA RANALLI, NATHAN SMITH, and BENJAMIN MCCARTY. "MINIMAL KNOTTING NUMBERS." Journal of Knot Theory and Its Ramifications 18, no. 08 (August 2009): 1159–73. http://dx.doi.org/10.1142/s0218216509007373.

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This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

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33

HART, JAMES B., LORI RAFTER, and CONSTANTINE TSINAKIS. "THE STRUCTURE OF COMMUTATIVE RESIDUATED LATTICES." International Journal of Algebra and Computation 12, no. 04 (August 2002): 509–24. http://dx.doi.org/10.1142/s0218196702001048.

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A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, ·, e) is a commutative monoid, (L, ∧, ∨) is a lattice, and the operation → satisfies the equivalences [Formula: see text] for a, b, c ∊ L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19–22]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an "ideal variety" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4].

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34

A. Edison, Enaibe, Akpata Erhieyovwe, and Osafile Omosede. "A VARIATIONAL THEORY OF QUASI-PARTICLES IN A 3D N x N x N CUBIC LATTICE." JOURNAL OF ADVANCES IN PHYSICS 5, no. 1 (August 2, 2014): 712–25. http://dx.doi.org/10.24297/jap.v5i1.6101.

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The single-band Hubbard Hamiltonian study faces a serious limitation and difficulty as we move away from finite - size lattices to larger N - dimensional lattices. Thus there is the needto develop the means of overcoming the finite - size lattice defects as we pass on to a higher dimension.In this work, a quantitative approximation to the one-band Hubbard model is presented using a variational analytic approach. The goal of this work, therefore, is to explore quantitatively the lowest ground-state energy and the pairing correlations in 3D N x N x N lattices of the Hubbard model. We developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationallythe correlation of two interacting elections on a three-dimensional cubic lattice. We also showed primarily how to derive possible electronic states available for several even and odd3D lattices, although, this work places more emphasis on a 3D 5 x 5 x 5 lattice. The results emerging from our present study compared favourablywith the results of Gutzwillervariational approach (GVA) and correlated variational approach (CVA), at thelarge limit of the Coulomb interaction strength (U/4t). It is revealed in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.

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35

Batson, Scott C. "The linear transformation that relates the canonical and coefficient embeddings of ideals in cyclotomic integer rings." International Journal of Number Theory 13, no. 09 (September 20, 2017): 2277–97. http://dx.doi.org/10.1142/s1793042117501251.

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The geometric embedding of an ideal in the algebraic integer ring of some number field is called an ideal lattice. Ideal lattices and the shortest vector problem (SVP) are at the core of many recent developments in lattice-based cryptography. We utilize the matrix of the linear transformation that relates two commonly used geometric embeddings to provide novel results concerning the equivalence of the SVP in these ideal lattices arising from rings of cyclotomic integers.

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36

Boutonnet, Rémi, and Cyril Houdayer. "Stationary characters on lattices of semisimple Lie groups." Publications mathématiques de l'IHÉS 133, no. 1 (March 2, 2021): 1–46. http://dx.doi.org/10.1007/s10240-021-00122-8.

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AbstractWe show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$ Γ < G , the left regular representation $\lambda _{\Gamma }$ λ Γ is weakly contained in any weakly mixing representation $\pi $ π . We prove that for any such irreducible lattice $\Gamma < G$ Γ < G , any Uniformly Recurrent Subgroup (URS) of $\Gamma $ Γ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$ Γ < G . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

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37

Booth, G. L., Q. N. Petersen, and S. Veldsman. "Lattices of Radicals of Ω-Groups." Algebra Colloquium 13, no. 03 (September 2006): 381–404. http://dx.doi.org/10.1142/s1005386706000332.

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Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

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38

Bezhanishvili, Guram, and Patrick J. Morandi. "Profinite Heyting Algebras and Profinite Completions of Heyting Algebras." gmj 16, no. 1 (March 2009): 29–47. http://dx.doi.org/10.1515/gmj.2009.29.

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Abstract This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.

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39

Longstaff, W. E., J. B. Nation, and Oreste Panaia. "Abstract reflexive sublattices and completely distributive collapsibility." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 245–60. http://dx.doi.org/10.1017/s0004972700032226.

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There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice ℒ is completely distributive, then ℒ is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on ℒ such that ℒ/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

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Albu, Toma, and Mihai Iosif. "Modular C11 lattices and lattice preradicals." Journal of Algebra and Its Applications 16, no. 06 (April 12, 2017): 1750116. http://dx.doi.org/10.1142/s021949881750116x.

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This paper deals with properties of modular [Formula: see text] lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

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41

Kumar, Kamlesh. "Lattice Theory of Fourth Order Elastic Constants of Primitive Lattices." Bulletin of Pure & Applied Sciences- Physics 40d, no. 2 (2021): 129–31. http://dx.doi.org/10.5958/2320-3218.2021.00020.8.

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42

McCune, W., and R. Padmanabhan. "Single identities for lattice theory and for weakly associative lattices." Algebra Universalis 36, no. 4 (December 1996): 436–49. http://dx.doi.org/10.1007/bf01233914.

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43

Goyal, S. C., and K. Kumar. "Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices." physica status solidi (b) 131, no. 2 (October 1, 1985): 451–57. http://dx.doi.org/10.1002/pssb.2221310206.

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44

DOMAGALSKA, P., and E. R. PUCZYŁOWSKI. "DIMENSION MODULES AND MODULAR LATTICES." Journal of Algebra and Its Applications 11, no. 05 (September 26, 2012): 1250082. http://dx.doi.org/10.1142/s021949881250082x.

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A module M is called a dimension module if the Goldie (uniform) dimension satisfies the formula u(A + B) + u(A ∩ B) = u(A) + u(B) for arbitrary submodules A, B of M. Dimension modules and related notions were studied by several authors. In this paper, we study them in a more general context of modular lattices with 0 to which the notion of dimension modules can be extended in an obvious way. Some constructions available in the lattice theory framework make it possible to identify several new aspects concerning the nature of dimension lattices and modules as well as to describe a number of related properties. In particular we find a lattice which can be used to test whether a given lattice or a module satisfies the studied properties. Most of the results are obtained for lattices and then they are applied to modules. However the examples are given, when possible, in the more restrictive case of modules.

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Wang, Zhangjun, and Zili Chen. "Applications for Unbounded Convergences in Banach Lattices." Fractal and Fractional 6, no. 4 (April 1, 2022): 199. http://dx.doi.org/10.3390/fractalfract6040199.

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Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices.

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Pliev, M. A. "Каждая латеральная полоса является ядром положительного ортогонально аддитивного оператора." Владикавказский математический журнал, no. 4 (December 23, 2021): 115–18. http://dx.doi.org/10.46698/e4075-8887-4097-s.

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{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

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47

Aichinger, Erhard. "Congruence lattices forcing nilpotency." Journal of Algebra and Its Applications 17, no. 02 (January 23, 2018): 1850033. http://dx.doi.org/10.1142/s0219498818500330.

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Given a lattice [Formula: see text] and a class [Formula: see text] of algebraic structures, we say that [Formula: see text] forces nilpotency in [Formula: see text] if every algebra [Formula: see text] whose congruence lattice [Formula: see text] is isomorphic to [Formula: see text] is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.

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Abuhlail, Jawad, and Christian Lomp. "On topological lattices and their applications to module theory." Journal of Algebra and Its Applications 15, no. 03 (January 27, 2016): 1650046. http://dx.doi.org/10.1142/s0219498816500468.

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Yassemi’s “second submodules” are dualized and properties of its spectrum are studied. This is done by moving the ring theoretical setting to a lattice theoretical one and by introducing the notion of a (strongly) topological lattice [Formula: see text] with respect to a proper subset [Formula: see text] of [Formula: see text] We investigate and characterize (strongly) topological lattices in general in order to apply it to modules over associative unital rings. Given a non-zero left [Formula: see text]-module [Formula: see text] we introduce and investigate the spectrum [Formula: see text] of first submodules of [Formula: see text] as a dual notion of Yassemi’s second submodules. We topologize [Formula: see text] and investigate the algebraic properties of [Formula: see text] by passing to the topological properties of the associated space.

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Kadji, Albert, Celestin Lele, Jean B. Nganou, and Marcel Tonga. "Folding Theory Applied to Residuated Lattices." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/428940.

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Residuated lattices play an important role in the study of fuzzy logic based ont-norms. In this paper, we introduce some notions ofn-fold filters in residuated lattices, study the relations among them, and compare them with prime, maximal and primary, filters. This work generalizes existing results in BL-algebras and residuated lattices, most notably the works of Lele et al., Motamed et al., Haveski et al., Borzooei et al., Van Gasse et al., Kondo et al., Turunen et al., and Borumand Saeid et al., we draw diagrams summarizing the relations between different types ofn-fold filters andn-fold residuated lattices.

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Woo, C. H., and W. Frank. "A Theory of Void Lattices." Materials Science Forum 15-18 (January 1987): 875–80. http://dx.doi.org/10.4028/www.scientific.net/msf.15-18.875.

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