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Journal articles on the topic 'Complete lattice'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Sangalli, Arturo A. L. "Lattices of fuzzy objects." International Journal of Mathematics and Mathematical Sciences 19, no. 4 (1996): 759–66. http://dx.doi.org/10.1155/s0161171296001056.

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The collection of fuzzy subsets of a setXforms a complete lattice that extends the complete lattice𝒫(X)of crisp subsets ofX. In this paper, we interpret this extension as a special case of the fuzzification of an arbitrary complete latticeA. We show how to construct a complete latticeF(A,L)theL-fuzzificatio ofA, whereLis the valuation lattice that extendsAwhile preserving all suprema and infima. The fuzzy objects inF(A,L)may be interpreted as the sup-preserving maps fromAto the dual ofL. In particular, each complete lattice coincides with its2-fuzzification, where2is the twoelement latti

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2

Pastijn, F., and P. G. Trotter. "Complete Congruences on Lattices of Varieties and of Pseudovarieties." International Journal of Algebra and Computation 08, no. 02 (1998): 171–201. http://dx.doi.org/10.1142/s0218196798000107.

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Three methods for the construction of all complete congruences on the lattice Lv( V ) of subvarieties of a variety V are introduced. It is shown that there exists an order preserving embedding of the lattice of complete congruences on the lattice Lp( P ) of all subpseudovarieties of a given pseudovariety P into the direct product of the lattices of complete congruences on lattices of subvarieties of varieties generated by members of P; thus there are methods for constructing all complete congruences on Lp( P ). By way of application, 2ℵ0 complete congruences and complete endomorphisms are cons

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3

FREESE, R., G. GRÄTZE, and E. T. SCHMIDT. "ON COMPLETE CONGRUENCE LATTICES OF COMPLETE MODULAR LATTICES." International Journal of Algebra and Computation 01, no. 02 (1991): 147–60. http://dx.doi.org/10.1142/s0218196791000080.

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The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In 1988, the second author announced the converse: every complete lattice L can be represented as the lattice of complete congruence relations of some complete lattice K. In this paper we improve this result by showing that K can be chosen to be a complete modular lattice.

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4

Yu, Bin, Qingguo Li, and Huanrong Wu. "A new view of relationship between atomic posets and complete (algebraic) lattices." Open Mathematics 15, no. 1 (2017): 238–51. http://dx.doi.org/10.1515/math-2017-0027.

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AbstractIn the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.

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5

Jakubík, Ján. "Complete retract mappings of a complete lattice ordered group." Czechoslovak Mathematical Journal 43, no. 2 (1993): 309–18. http://dx.doi.org/10.21136/cmj.1993.128396.

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6

Dai, Songsong. "Rough Approximation Operators on a Complete Orthomodular Lattice." Axioms 10, no. 3 (2021): 164. http://dx.doi.org/10.3390/axioms10030164.

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This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

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7

Bessonov, Yu E., and A. A. Dobrynin. "Lattice complete graphs." Journal of Applied and Industrial Mathematics 11, no. 4 (2017): 481–85. http://dx.doi.org/10.1134/s1990478917040032.

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8

Booth, G. L., Q. N. Petersen та S. Veldsman. "Lattices of Radicals of Ω-Groups". Algebra Colloquium 13, № 03 (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 genera

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9

Boyu, Li. "All retraction operators on a complete lattice form a complete lattice." Acta Mathematica Sinica 7, no. 3 (1991): 247–51. http://dx.doi.org/10.1007/bf02583001.

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10

Zhang, Zhongxi, Qingguo Li, and Nan Zhang. "m-Algebraic lattices in formal concept analysis." Mathematical Structures in Computer Science 29, no. 10 (2019): 1556–74. http://dx.doi.org/10.1017/s0960129519000124.

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AbstractThe notion of an m-algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m-approximable concept on each context. The m-approximable concept lattice derived from the notion is an m-algebraic lattice, and conversely, every m-algebraic lattice is isomorphic to an m-ap

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11

Jakubík, Ján. "Affine completeness of complete lattice ordered groups." Czechoslovak Mathematical Journal 45, no. 3 (1995): 571–76. http://dx.doi.org/10.21136/cmj.1995.128545.

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12

SWAMY, U. M., and B. VENKATESWARLU. "IRREDUCIBLE ELEMENTS IN ALGEBRAIC LATTICES." International Journal of Algebra and Computation 20, no. 08 (2010): 969–75. http://dx.doi.org/10.1142/s0218196710005984.

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α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements,

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13

Morales, Marcel, and Apostolos Thoma. "Complete intersection lattice ideals." Journal of Algebra 284, no. 2 (2005): 755–70. http://dx.doi.org/10.1016/j.jalgebra.2004.10.011.

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14

Herbut, Fedor. "Latent Complete-Lattice Structure of Hilbert-Space Projectors." Quanta 8, no. 1 (2019): 1. http://dx.doi.org/10.12743/quanta.v8i1.85.

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To uncover the hidden complete-lattice structure of Hilbert-space projectors, which is not seen by the operator operations and relations (algebraically), resort is taken to the ranges of projectors (to subspaces—to geometry). Taking the range of a projector is completed into a bijection of all projectors onto all subspaces of any finite or countably infinite dimensional Hilbert space. As a second step, this basic bijection is upgraded into an isomorphism of partially ordered sets utilizing the sub-projector relation on the one hand, and the subspace relation on the other. As a third and final

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15

Luo-Shan, Xu. "Construct fuzzy lattices from a given symmetric complete lattice." Fuzzy Sets and Systems 66, no. 3 (1994): 357–62. http://dx.doi.org/10.1016/0165-0114(94)90103-1.

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16

Georgiou, D. N., and A. C. Megaritis. "The quasi Scott (Lawson) topology and q-continuous (q-algebraic) complete lattices." Filomat 29, no. 1 (2015): 193–207. http://dx.doi.org/10.2298/fil1501193g.

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Let L be a complete lattice. On L we define the so called quasi Scott topology, denoted by ?qSc. This topology is always larger than or equal to the Scott topology and smaller than or equal to the strong Scott topology. Results concerning the above topology are given. Also, we introduce and investigate the notions of q-continuous and q-algebraic complete lattices. Finally, we give and examine the quasi Lawson topology on a complete lattice.

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17

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 (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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18

Pennec, Yan, Bahram DjafariRouhani, EH ElBoudouti, et al. "Simultaneous Existence of Phononic and Photonic Band Gaps in Phoxonic Crystal Slabs." Optics express 18, no. 13 (2010): 14301–10. https://doi.org/10.1364/OE.18.014301.

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We discuss the simultaneous existence of phononic and photonic band gaps in a periodic array of holes drilled in a Si membrane. We investigate in detail both the centered square lattice and the boron nitride (BN) lattice with two atoms per unit cell which include the simple square, triangular and honeycomb lattices as particular cases. We show that complete phononic and photonic band gaps can be obtained from the honeycomb lattice as well as BN lattices close to honeycomb. Otherwise, all investigated structures present the possibility of a complete phononic gap together with a photonic band ga

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19

ZHANG, WENFENG, and XIAOQUAN XU. "A completion-invariant extension of the concept of meet continuous lattices." Mathematical Structures in Computer Science 27, no. 4 (2015): 530–39. http://dx.doi.org/10.1017/s0960129515000213.

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In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

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20

Jakubík, Ján. "On complete lattice ordered groups with strong units." Czechoslovak Mathematical Journal 46, no. 2 (1996): 221–30. http://dx.doi.org/10.21136/cmj.1996.127285.

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21

Grätzer, G., and H. Lakser. "On congruence lattices of m-complete lattices." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 52, no. 1 (1992): 57–87. http://dx.doi.org/10.1017/s1446788700032869.

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AbstractThe lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruen

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22

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 (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 latti

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23

Semenov, Alexei, and Sergei Soprunov. "Automorphisms and Definability (of Reducts) for Upward Complete Structures." Mathematics 10, no. 20 (2022): 3748. http://dx.doi.org/10.3390/math10203748.

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The Svenonius theorem establishes the correspondence between definability of relations in a countable structure and automorphism groups of these relations in extensions of the structure. This may help in finding a description of the lattice constituted by all definability spaces (reducts) of the original structure. Results on definability lattices were previously obtained only for ω-categorical structures with finite signature. In our work, we introduce the concept of an upward complete structure and define the upward completion of a structure. For upward complete structures, the Galois corres

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24

Tallee Kakeu, Ariane G., Lutz Strüngmann, Blaise B. Koguep Njionou та Celestin Lele. "ℒ-fuzzy Annihilators in Residuated Lattices". Mathematica Slovaca 73, № 6 (2023): 1359–72. http://dx.doi.org/10.1515/ms-2023-0098.

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ABSTRACT In this paper, we provide a new characterization of ℒ -fuzzy ideals of residuated lattices, which allows us to describe ℒ -fuzzy ideals generated by ℒ -fuzzy sets. Thanks to the latter, we endow the lattice of ℒ -fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ -fuzzy annihilator of an ℒ -fuzzy subset of a residuated lattice with respect to an ℒ -fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ -fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define

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Dissanayake, S. E., and K. A. I. L. Wijewardena Gamalath. "Simulation of Two Dimensional Photonic Band Gaps." International Letters of Chemistry, Physics and Astronomy 24 (December 2013): 58–88. http://dx.doi.org/10.18052/www.scipress.com/ilcpa.24.58.

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The plane wave expansion method was implemented in modelling and simulating the band structures of two dimensional photonic crystals with square, triangular and honeycomb lattices with circular, square and hexagonal dielectric rods and air holes. Complete band gaps were obtained for square lattice of square GaAs rods and honeycomb lattice of circular and hexagonal GaAs rods as well as triangular lattice of circular and hexagonal air holes in GaAs whereas square lattice of square or circular air holes in a dielectric medium ε = 18 gave complete band gaps. The variation of these band gaps with d

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Dissanayake, S. E., and K. A. I. L. Wijewardena Gamalath. "Simulation of Two Dimensional Photonic Band Gaps." International Letters of Chemistry, Physics and Astronomy 24 (December 26, 2013): 58–88. http://dx.doi.org/10.56431/p-41l177.

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The plane wave expansion method was implemented in modelling and simulating the band structures of two dimensional photonic crystals with square, triangular and honeycomb lattices with circular, square and hexagonal dielectric rods and air holes. Complete band gaps were obtained for square lattice of square GaAs rods and honeycomb lattice of circular and hexagonal GaAs rods as well as triangular lattice of circular and hexagonal air holes in GaAs whereas square lattice of square or circular air holes in a dielectric medium ε = 18 gave complete band gaps. The variation of these band gaps with d

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27

Jakubik, Jan. "Complete Distributivity of Lattice Ordered Groups and of Vector Lattices." Czechoslovak Mathematical Journal 51, no. 4 (2001): 889–96. http://dx.doi.org/10.1023/a:1013781300217.

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28

Catterall, Simon, and Aarti Veernala. "A complete lattice technicolor model." International Journal of Modern Physics A 29, no. 25 (2014): 1445002. http://dx.doi.org/10.1142/s0217751x1445002x.

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We construct a lattice gauge theory using reduced staggered fermions and gauge fields which provides a nonperturbative realization of a complete technicolor model; one which treats both strong and weakly coupled gauge sectors on an equal footing. We show that the model is capable of developing a Higgs phase at nonzero lattice spacing via the formation of fermion condensates. We show further that while the broken symmetry associated with this phase has a vector character in the lattice theory it is realized as an axial symmetry in the continuum limit in agreement with the Vafa–Witten theorem. W

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NURAKUNOV, A. M. "UNREASONABLE LATTICES OF QUASIVARIETIES." International Journal of Algebra and Computation 22, no. 03 (2012): 1250006. http://dx.doi.org/10.1142/s0218196711006728.

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A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set [Formula: see text] of all subquasivarieties of a quasivariety [Formula: see text] forms a complete lattice under inclusion. A lattice isomorphic to [Formula: see text] for some quasivariety [Formula: see text] is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algor

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30

Jakubík, Ján. "On disjoint subsets of a complete lattice ordered group." Časopis pro pěstování matematiky 115, no. 2 (1990): 165–70. http://dx.doi.org/10.21136/cpm.1990.108360.

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31

SEBASTIAN, SABU, and T. V. RAMAKRISHNAN. "MULTI-FUZZY EXTENSIONS OF FUNCTIONS." Advances in Adaptive Data Analysis 03, no. 03 (2011): 339–50. http://dx.doi.org/10.1142/s1793536911000714.

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In this paper, we study various properties of multi-fuzzy extensions of crisp functions using order homomorphisms, complete lattice homomorphisms, L-fuzzy lattices, and strong L-fuzzy lattices as bridge functions.

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32

Valverde-Albacete, Francisco, and Carmen Peláez-Moreno. "The Singular Value Decomposition over Completed Idempotent Semifields." Mathematics 8, no. 9 (2020): 1577. http://dx.doi.org/10.3390/math8091577.

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In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a

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33

Auinger, Karl. "Semigroups with atomistic congruence lattices." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 52, no. 1 (1992): 88–102. http://dx.doi.org/10.1017/s1446788700032870.

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AbstractIn this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

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34

Arworn, Srichan. "Characterizations of Complete Sublattices of a Given Complete Lattice." Southeast Asian Bulletin of Mathematics 25, no. 2 (2001): 191–200. http://dx.doi.org/10.1007/s10012-001-0191-1.

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35

Auinger, Karl. "The congruence lattice of a combinatorial strict inverse semigroup." Proceedings of the Edinburgh Mathematical Society 37, no. 1 (1994): 25–37. http://dx.doi.org/10.1017/s0013091500018654.

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36

Rasouli, Saeed. "Galois connection of stabilizers in residuated lattices." Filomat 34, no. 4 (2020): 1223–39. http://dx.doi.org/10.2298/fil2004223r.

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The paper is devoted to introduce the notions of some types of stabilizers in non-commutative residuated lattices and to investigate their properties. We establish a connection between (contravariant) Galois connection and stabilizers of a residuated lattices. If A is a residuated lattice and F be a filter of A, we show that the set of all stabilizers relative to F of a same type forms a complete lattice. Furthermore, we prove that ST - F?l, ST - Fl and ST - Fs are pseudocomplemented lattices.

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Rasouli, Saeed. "Galois connection of stabilizers in residuated lattices." Filomat 34, no. 4 (2020): 1223–39. http://dx.doi.org/10.2298/fil2004223r.

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The paper is devoted to introduce the notions of some types of stabilizers in non-commutative residuated lattices and to investigate their properties. We establish a connection between (contravariant) Galois connection and stabilizers of a residuated lattices. If A is a residuated lattice and F be a filter of A, we show that the set of all stabilizers relative to F of a same type forms a complete lattice. Furthermore, we prove that ST - F?l, ST - Fl and ST - Fs are pseudocomplemented lattices.

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Li, He, Lingjie Li, Haozhang Zhong, Hanxuan Mo, and Mengyuan Gu. "Hierarchical lattice: Design strategy and topology characterization." Advances in Mechanical Engineering 15, no. 6 (2023): 168781322311796. http://dx.doi.org/10.1177/16878132231179623.

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The structure-material integrated design is an art-of-state concept and be enabled by additive manufacturing. The lattice material is classified into structure as well as material because mechanical properties are determined by its topology. However, the lack of a flexible design strategy hinders the lattice achieve the structure-material integrated material candidate. This work suggests the strut-nested based strategies to effectively conduct the hierarchical lattice design. The strut in the larger-scale lattice can be replaced by the smaller-scale lattice structure through the rotation, stre

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Belohlavek, Radim, and Vilem Vychodil. "Fuzzy Concept Lattices Constrained by Hedges." Journal of Advanced Computational Intelligence and Intelligent Informatics 11, no. 6 (2007): 536–45. http://dx.doi.org/10.20965/jaciii.2007.p0536.

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We study concept lattices constrained by hedges. The principal aim is to control, in a parameterical way, the size of concept lattices, i.e. the number of conceptual clusters extracted from data. The paper presents theoretical insight, comments, and examples. We introduce new, parameterized, concept-forming operators and study their properties. We obtain an axiomatic characterization of the concept-forming operators. Then, we show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. Thes

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GHORBANI, SHOKOOFEH. "INTUITIONISTIC FUZZY FILTERS OF RESIDUATED LATTICES." New Mathematics and Natural Computation 07, no. 03 (2011): 499–513. http://dx.doi.org/10.1142/s1793005711002049.

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In this paper, the concept of intuitionistic fuzzy sets is applied to residuated lattices. The notion of intuitionistic fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of intuitionistic fuzzy filters are obtained. We show that the set of all the intuitionistic fuzzy filters of a residuated lattice forms a complete lattice and we find the distributive sublattices of it. Finally, the correspondence theorem for intuitionistic fuzzy filters is established.

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Chajda, Ivan, and Helmut Länger. "When does a semiring become a residuated lattice?" Asian-European Journal of Mathematics 09, no. 04 (2016): 1650088. http://dx.doi.org/10.1142/s1793557116500881.

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It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is o

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MAŇUCH, JÁN, and DAYA RAM GAUR. "FITTING PROTEIN CHAINS TO CUBIC LATTICE IS NP-COMPLETE." Journal of Bioinformatics and Computational Biology 06, no. 01 (2008): 93–106. http://dx.doi.org/10.1142/s0219720008003308.

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It is known that folding a protein chain into a cubic lattice is an NP-complete problem. We consider a seemingly easier problem: given a three-dimensional (3D) fold of a protein chain (coordinates of its C α atoms), we want to find the closest lattice approximation of this fold. This problem has been studied under names such as "lattice approximation of a protein chain", "the protein chain fitting problem", and "building of protein lattice models". We show that this problem is NP-complete for the cubic lattice with side close to 3.8 Å and coordinate root mean square deviation.

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43

Laterveer, Robert. "Weighted complete intersections and lattice points." Mathematische Zeitschrift 218, no. 1 (1995): 213–18. http://dx.doi.org/10.1007/bf02571899.

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44

Chikatamarla, Shyam S., and Iliya V. Karlin. "Complete Galilean invariant lattice Boltzmann models." Computer Physics Communications 179, no. 1-3 (2008): 140–43. http://dx.doi.org/10.1016/j.cpc.2008.01.037.

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45

Kitano, Yasuyuki, and Masaki Takata. "Coincidence-site-lattice-pattern (CSL-pattern) of 70.5°/[110] boundary of the 6H-type layer structure." Proceedings, annual meeting, Electron Microscopy Society of America 48, no. 4 (1990): 356–57. http://dx.doi.org/10.1017/s0424820100174916.

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The most useful and intuitive model may be a CSL-model to analyze boundary structures. In order to apply the CSL-model to layer structures, we have proposed to use ‘lattice point’ in a wide sence and to add extra lattice points to the Bravais lattice points when interpenetrating(IP)-lattices are drawn. These lattice points will be called ‘extended lattice points’. It is well known that a layer structure is built up with (almost) identical layers stacking on the top of the others with a cirtain amount of shift in a direction perpendicular to the stacking. Each layer consists of one or more atom

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46

Wójtowicz, Marek. "The lattice-isometric copies ofℓ∞(Γ)in quotients of Banach lattices". International Journal of Mathematics and Mathematical Sciences 2003, № 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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47

Bonet, José, Ben de Pagter, and Werner J. Ricker. "Mean ergodic operators and reflexive Fréchet lattices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 5 (2011): 897–920. http://dx.doi.org/10.1017/s0308210510000314.

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Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every p

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48

Rao, G. C., and Venugopalam Undurthi. "Complete almost distributive lattices." Asian-European Journal of Mathematics 07, no. 03 (2014): 1450052. http://dx.doi.org/10.1142/s1793557114500521.

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49

Jenča, Gejza. "The block structure of complete lattice ordered effect algebras." Journal of the Australian Mathematical Society 83, no. 2 (2007): 181–216. http://dx.doi.org/10.1017/s1446788700036867.

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AbstractWe prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).

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50

Rana, Pravanjan Kumar, and Sarmad Hossain. "A study of complete lattices of covering spaces." Annals of Mathematics and Computer Science 18 (October 1, 2023): 1–5. http://dx.doi.org/10.56947/amcs.v18.185.

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Let C(X) denote the set of all covering spaces (X', x', p) of (X,x) where (X,x) are path connected,locally path connected and semilocally simply connected pointed topological spaces. In [1], it is shown that (C(X),≥) is a lattice with respect to the partial order relation ≥. Also (C(X),≥) is a modular, bounded and complete lattice when π(X,x) is abelian. In this paper, we will study some properties of the complete lattice (C(X), ≥).

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