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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
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Semimodular Lattices
- Theory and Applications
- Manfred Stern
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In Semimodular Lattices: Theory and Applications Manfred Stern uses successive generalizations of distributive and modular lattices to outline the development of semimodular lattices from Boolean algebras. He focuses on the important theory of semimodularity, its many ramifications, and its applications in discrete mathematics, combinatorics, and algebra. The book surveys and analyzes Garrett Birkhoff's concept of semimodularity and the various related concepts in lattice theory, and it presents theoretical results as well as applications in discrete mathematics group theory and universal algebra. The author also deals with lattices that are 'close' to semimodularity or can be combined with semimodularity, e.g. supersolvable, admissible, consistent, strong, and balanced lattices. Researchers in lattice theory, discrete mathematics, combinatorics, and algebra will find this book invaluable.
5 - The Covering Graph
- Manfred Stern
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Summary
Diagrams and Covering Graphs
Summary. We make some general remarks on the Hasse diagram and the covering graph of a finite poset. Then we turn to the question of orientations and reorientations of a covering graph. Birkhoff [1948] asked for necessary and sufficient conditions on a lattice L in order that every lattice M whose covering graph is isomorphic with the covering graph of L be lattice isomorphic to L. Solutions to this problem will be given in the subsequent sections.
The Hasse diagram (briefly: the diagram) of a finite poset P = (P, ≤) is an oriented graph with the circles of P as its vertices and an edge x → y if y covers x in P. Usually the arrows on the edges are omitted and the graph is arranged so that all edges point upwards on the page. The diagram of a finite poset is the most common tool for representing the poset graphically. For example, the usual diagram of the lattice 23 of all subsets of a three-element set, ordered by set inclusion, is shown in Figure 1.3(c) (Section 1.2). The diagram of a finite poset P determines P up to isomorphism. The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph. In this chapter we shall have a closer look at the covering graph of certain lattices.
For a finite poset P, its covering graph G(P) was introduced in Section 1.9. As for diagrams, it is common to identify a pictorial representation of the covering graph with the covering graph itself.
Preface
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This book aims at giving a survey of semimodularity and related concepts in lattice theory as well as presenting a number of applications. The book may be regarded as a supplement to certain aspects of vol. 26 (Theory of Matroids), vol. 29 (Combinatorial Geometries), and vol. 40 (Matroid Applications) of this encyclopedia.
Classically semimodular lattices arose out of certain closure operators satisfying what is now usually called the Steinitz–Mac Lane exchange property. Inspired by the matroid concept introduced in 1935 by Hassler Whitney in a paper entitled “On the abstract properties of linear dependence,” Garrett Birkhoff isolated the concept of semimodularity in lattice theory. Matroids are related to geometric lattices, that is, to semimodular atomistic lattices of finite length. The theory of geometric lattices was not foreshadowed in Dedekind's work on modular lattices. Geometric lattices were the first class of semimodular lattices to be systematically investigated.
The theory was developed in the thirties by Birkhoff, Wilcox, Mac Lane, and others. In the early forties Dilworth discovered locally distributive lattices, which turned out to be important new examples of semimodular lattices. These examples are the first cryptomorphic versions of what became later known in combinatorics as antimatroids. The name antimatroid hints at the fact that this combinatorial structure has properties that are very different from certain matroid properties. In particular, antimatroids have the so-called antiexchange property. While matroids abstract the notion of linear independence, antimatroids abstract the notion of Euclidean convexity. The theory was further developed by Dilworth, Crawley, Avann, and others in the fifties and sixties.
During the years 1935–55 and later many ramifications and applications of semimodularity were discovered.
Index
- Manfred Stern
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1 - From Boolean Algebras to Semimodular Lattices
- Manfred Stern
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Sources of Semimodularity
Summary. We briefly indicate some milestones in the general development of lattice theory. In particular, we outline the way leading from Boolean algebras to semimodular lattices. Most of the concepts mentioned in this section will be explained in more detail later. We also give a number of general references and monographs on lattice theory and its history.
Boolean Algebras and Distributive Lattices
Lattice theory evolved in the nineteenth century through the works of George Boole, Charles Saunders Peirce, and Ernst Schröder, and later in the works of Richard Dedekind, Garrett Birkho21, Oystein Ore, John von Neumann, and others during the first half of the twentieth century. Boole [1847] laid the foundation for the algebras named after him. Since then the more general distributive lattices have been investigated whose natural models are systems of sets. There are many monographs on Boolean algebras and their applications, such as Halmos [1963] and Sikorski [1964]. For the theory of distributive lattices we refer to the books by Grätzer [1971] and Balbes & Dwinger [1974].
Modular Lattices
Dedekind [1900] observed that the additive subgroups of a ring and the normal subgroups of a group form lattices in a natural way (which he called Dualgruppen) and that these lattices have a special property, which was later referred to as the modular law. Modularity is a consequence of distributivity, and Dedekind's observation gave rise to examples of nondistributive modular lattices.
Lattice theory became established in the 1930s due to the contributions of Garrett Birkhoff, Ore, Menger, von Neumann, Wilcox, and others.
Table of Notation
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9 - Congruence Semimodularity
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Summary
Semilattices
Summary. Algebras whose congruence lattices satisfy nontrivial lattice identities have been thoroughly studied during the last two decades. In particular, varieties have been investigated in which the congruence lattice of all algebras is distributive or modular. Although weaker than modularity, semimodularity is still an interesting property in this connection. Ore [1942] proved that sets have semimodular congruence lattices (in fact, these are matroid lattices). Here we describe some properties of congruence lattices of semilattices that were shown to be semimodular by Hall [1971].
An algebra S = (S: ·) with a binary operation · is called a semigroup if this operation is associative, that is, x · (y · z) = (x · y) · z holds for all x, y, z ∈ S. For simplification we briefly write ab for a · b. We shall identify S with S if there is no danger of confusion. A semilattice is a semigroup satisfying xy = yx (commutativity) and x2 = x (idempotency) for all x, y ∈ S. We may impose a partial order on S by defining x ≤ y if xy = x. Under this ordering, any two elements x, y ∈ S have a greatest lower bound, namely their product xy. Hence S is a (meet) semilattice.
An algebra A is said to be locally finite if every subalgebra of A generated by finitely many elements is finite. A variety ∨ is said to be locally finite if every algebra in ∨ is locally finite, or equivalently, every finitely generated algebra in ∨ is finite. Semilattices form a locally finite variety. We denote this variety by Semilattices.
4 - Supersolvable and Admissible Lattices; Consistent and Strong Lattices
- Manfred Stern
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3 - Conditions Related to Semimodularity, 0-Conditions, and Disjointness Properties
- Manfred Stern
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Summary
Mac Lane's Condition
Summary. In Chapter 2 we dealt with M-symmetry, a condition related to semimodularity in the sense that the two conditions are equivalent in lattices of finite length. Wilcox's concept of M-symmetry is one important approach to the question of replacing upper semimodularity by a condition that is nontrivially satisfied in lattices having continuous chains. Another approach is that of Mac Lane [1938], at which we cast a glance here. His condition will be given in two equivalent forms. We study the interrelationships between Mac Lane's condition, semimodularity, Birkhoff's condition, and M-symmetry and show, among other things, that these conditions are equivalent in upper continuous strongly atomic lattices.
In atomistic lattices, the Steinitz–Mac Lane exchange property [condition (EP)] is equivalent to the atomic covering property [condition (C)] and thus to upper semimodularity [condition (Sm)] (see Theorem 1.7.3). On the other hand, the conditions (Sm), (EP), and (C) are trivially satisfied in lattices with continuous chains. It was this criticism of (Sm) that led Wilcox [1938], [1939] to introduce M-symmetric lattices (cf. Chapter 2).
Mac Lane [1938] was led in his investigations to an “exchange axiom,” denoted by him as condition (E5), which is stronger than (Sm) and goes in a different direction than M-symmetry. Like M-symmetry, Mac Lane's condition does not involve any covering relation. Mac Lane's point of departure was the following condition due to Menger [1936] : If p is an atom and a, c are arbitrary elements, then p ≰ a ∨ c implies (a ∨ p) ∧ c = a ∧ c. Mac Lane called this condition Menger's exchange axiom and denoted it by (E3).
Frontmatter
- Manfred Stern
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6 - Semimodular Lattices of Finite Length
- Manfred Stern
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2 - M-Symmetric Lattices
- Manfred Stern
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Modular Pairs and Modular Elements
Summary. Modular pairs were defined in Section 1.2, and later several properties were given, including a characterization via certain mappings (see Section 1.9). Here we give another characterization in terms of forbidden pentagons and some consequences. We present the parallelogram law, which is an extension of the isomorphism theorem (Dedekind's transposition principle) for modular lattices.
Blyth & Janowitz [1972], Theorem 8.1, p. 72, provided the following characterization of modular pairs in terms of relative complements, that is, excluding certain pentagon sublattices.
Theorem 2.1.1 Let a, b be elements of a lattice L. Then a M b holds if and only if L does not possess a pentagon sublattice {a ∧ b,a,x, y, a ∨ x = a ∨ y} with x < y ≤ b (see Figure 2.1).
Proof. If a M b and there exists a sublattice of the indicated form, then x < y ≤ b and thus y = (x ∨ a) ∧ y = (x ∨ a) ∧ b ∧ y = x ∨(a ∧ b)∧ y = x ∧ y = x, a contradiction. If a M b fails, then we can find an element t < b such that t ∨ (a ∧ b) < (t ∨ a) ∧ b. Setting x = t ∨ (a ∧ b) and y = (t ∨ a) ∧ b, we get a ∧ b ≤ a ∧ y = a ∧[(t ∨ a) ∧ b] = a ∧ b and hence a ∧ b = a ∧ y = a ∧ x. Dually we get a ∨ x = a∨ y = a ∨ t. Thus we obtain a pentagon sublattice of the required form.
Master Reference List
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Contents
- Manfred Stern
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7 - Local Distributive
- Manfred Stern
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8 - Local Modularity
- Manfred Stern
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The Kurosh–Ore Replacement Property
Summary. The Kurosh–Ore theorem for finite decompositions in modular lattices was given in Section 1.6. In particular we have seen that modularity implies the Kurosh–Ore replacement property for meet decompositions (∧-KORP). On the other hand, the lattice N5 shows that the ∧-KORP (together with its dual, the ∨-KORP) does not even imply semimodularity. Thus the question arose how to characterize the Kurosh–Ore replacement property in general and, in particular, in the semimodular case. The pertinent fundamental results are due to Dilworth and Crawley. In this section we have a brief look at the characterization of the ∧-KORP for strongly atomic algebraic lattices. In the following section we turn to the semimodular case.
The equivalence of the ∧-KORP, Crawley's condition (Cr*), and dual consistency for lattices of finite length was mentioned in Theorem 4.5.1.
From Section 1.8 we recall the definition of completely meet-irreducible elements and the fact that in an algebraic lattice every element is a meet of completely meet-irreducible elements, that is, infinite meet decompositions exist (cf. Theorem 1.8.1). We also recall that if an algebraic lattice is strongly atomic, then any meet-irreducible element is completely meet-irreducible, that is, the two concepts are identical. Moreover, Crawley [1961] proved the existence of irredundant meet decompositions (cf. Theorem 1.8.2).
In Section 1.8 the ∧-KORP was defined for complete lattices. Crawley's condition (Cr*) (defined in Section 4.5) can also be formulated for strongly atomic algebraic lattices. Finally let us state that we may define consistency for strongly dually atomic dually algebraic lattices as Kung [1985] did for lattices of finite length (cf. Section 4.5). In a dual way we define dual consistency for strongly atomic algebraic lattices.