DOCSLIB.ORG

Free Lattices Department of Mathematics University of Hawaii Honolulu, Hawaii 96822 Ralph@@Math.Hawaii.Edu MFF, Charles Universi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Free Lattices Department of Mathematics University of Hawaii Honolulu, Hawaii 96822 Ralph@@Math.Hawaii.Edu MFF, Charles Universi Free Lattices Ralph Freese Department of Mathematics University of Hawaii Honolulu, Hawaii 96822 ralph@@math.hawaii.edu Jaroslav Jeˇzek MFF, Charles University Sokolovsk´a83 18600 Praha 8 Czech Republic jezek@@karlin.mff.cuni.cz J. B. Nation Department of Mathematics University of Hawaii Honolulu, Hawaii 96822 jb@@math.hawaii.edu 1991 Mathematics Subject Classification. Primary 06B25; Secondary 06–04, 68Q25, 06A06, 06B20 This research supported by NSF Grants DMS–9204481 and DMS–9122011. Contents Introduction 1 Chapter I. Whitman’s Solution to the Word Problem 9 1. Basic Concepts from Lattice Theory 9 2. Free Lattices 13 3. Canonical Form 20 4. Continuity 25 5. Fixed Point Free Polynomials and Incompleteness 26 6. Sublattices of Free Lattices 30 Chapter II. Bounded Homomorphisms and Related Concepts 35 1. Bounded Homomorphisms 35 2. Continuity 48 3. Doubling and Congruences on a Finite Lattice 49 4. A Refinement of the D Relation 59 5. Semidistributive Lattices 64 6. Splitting Lattices 72 7. Day’s Theorem: Free Lattices are Weakly Atomic 81 8. Applications to Congruence Varieties 85 Chapter III. Covers in Free Lattices 89 1. Elementary Theorems on Covers in FL(X) 89 2. J–closed Sets and the Standard Epimorphism 94 3. Finite Lower Bounded Lattices 99 4. The Lattice L ⊔ (w) 102 5. Syntactic Algorithms 105 6. Examples 110 7. Connected Components and the Bottom of FL(X) 115 Chapter IV. Day’s Theorem Revisited 119 Chapter V. Sublattices of Free Lattices and Projective Lattices 125 1. Projective Lattices 125 2. The Free Lattice Generated by an Ordered Set 136 3. Finite Sublattices of Free Lattices 143 4. Related Topics 166 iii iv CONTENTS 5. Finite Subdirectly Irreducible Sublattices of Free Lattices 170 6. Summary 173 CONTENTS v Chapter VI. Totally Atomic Elements 179 1. Characterization 179 2. Canonical Form of Kappa of a Totally Atomic Element 187 3. The Role of Totally Atomic Elements 192 Chapter VII. Finite Intervals and Connected Components 199 1. Chains of Covers 199 2. Finite Intervals 204 3. Three Element Intervals 207 4. Connected Components 212 Chapter VIII. Singular and Semisingular Elements 225 1. Semisingular Elements 225 2. Singular Elements 233 Chapter IX. Tschantz’s Theorem and Maximal Chains 237 1. Tschantz’s Theorem 237 2. Coverless Elements 243 3. Maximal Chains 244 Chapter X. Infinite Intervals 249 1. Tschantz Triples 249 2. Join Irreducible Elements that are not Canonical Joinands 262 3. Splittings of a Free Lattice 264 Chapter XI. Term Rewrite Systems and Varieties of Lattices 271 1. Term Rewrite Systems 272 2. NoACTRSforLatticeTheory 275 3. An Extension 278 4. The Variety Generated by L ⊔ (w) 280 5. A Lattice Variety with AC TRS 288 6. More Varieties with AC TRS 295 Open Problems 301 Bibliography 303 List of Symbols 311 Index 313 Introduction This monograph covers the fascinating subject of free lattices. The arithmetic of free lattices is interesting and their structure is surpris- ingly complex. Although free lattices do have several nice properties, some reminiscent of properties of distributive lattices, discovering their nature is not easy. This book presents a thorough development of the structure of free lattices. A brief history. Although Richard Dedekind found the free mod- ular lattice on three generators in 1900, Thoralf Skolem showed the word problem for finitely presented lattices had a uniform solution in 1920, and Garrett Birkhoff showed the free lattice on three generators was infinite, substantial insight into the nature of free lattices really began with the two papers [132, 133] of Philip Whitman which ap- peared in 1941 and 1942. In these he solved the word problem for free lattices and showed that there was a unique shortest term representing each element of the free lattice, known as the canonical form. He also established a strong connection between this syntactic notion and the arithmetic of the lattice. He showed that the canonical form is the best way to represent an element in a very natural lattice theoretical sense. Using this interplay, he was able to prove several strong results about free lattices; for example, the free lattice on a countably infinite generating set is embeddable in the one on a three element generating set. In the 1950’s and 1960’s, R. Dean, B. J´onsson, F. Galvin, and J. Kiefer obtained further structural results about free lattices. One of their primary interests was sublattices of free lattices, particularly finite sublattices. As part of his solution to the word problem, Whitman had shown that free lattices satisfy an important property now known as Whitman’s condition, which states that the relation a b c d can hold in a free lattice only in one of the four trivial ways:∧ ≤ ∨ a c d, b c d, a b c, or a b d. ≤ ∨ ≤ ∨ ∧ ≤ ∧ ≤ J´onsson showed that free lattices and all their sublattices satisfy a weak form of distributivity known as semidistributivity. As part of 1 2 INTRODUCTION their attack on the problem of characterizing finite sublattices of free lattices, J´onsson and Kiefer showed that if a lattice satisfies Whitman’s condition and has a set of five elements, no four of which join above the fifth, then it is infinite. This implies that the breadth of a finite sublattice of a free lattice is at most four and that there is a nontrivial lattice equation true in all finite sublattices of a free lattice. Chapter I presents these and other classical results after first giving Whitman’s solution to the word problem and other basic results. It also shows that free lattices are continuous, not complete, and that free lattices have a unary polynomial which is fixed point free. The first chapter is suitable as an introduction to free lattices in, say, a course on lattice theory or algebra. In the paper by J´onsson and Kiefer it was conjectured that a finite lattice could be embedded into a free lattice if and only if it satisfies Whitman’s condition and is semidistributive. J´onsson worked on this conjecture, producing some unpublished notes that influenced further investigations by others. Ralph McKenzie’s important paper [99] uses J´onsson’s notes in the characterization of finite projective lattices. A great deal of work was put into J´onsson’s conjecture, as it was usually termed. J. B. Nation finally confirmed the conjecture in a deep paper [104] appearing in 1982. Nation’s proof is based on the approach de- veloped in J´onsson’s notes. This theorem appears in Chapter V, which also contains a characterization of projective lattices and other results on sublattices of free lattices. The notion of covering, one element being covered by another, de- noted a b, naturally leads to several intriguing questions and plays an extremely≺ important role in many of the deeper results on free lat- tices. Consequently, the theory of the covering relation in free lattices is well developed in this monograph. Whitman was the first to exhibit coverings in free lattices: he showed, for example, that the meet of all but one of the generators covers the meet of all of the generators. R. Dean, in unpublished work, showed that there were elements that covered no other element. When a b, we say a is covered by b and a is a lower cover of b. ≺ When a b (in any lattice), there is a unique maximal congruence relation, ψ(a,≺ b), not containing a, b . McKenzie [99] showed that the homomorphic image of the free latticeh i corresponding to this congruence relation is a finite, subdirectly irreducible lattice with the additional property that each element of this lattice has a least and a greatest preimage in the free lattice, i.e., the natural homomorphism is bounded. These lattices, which are called splitting lattices, and the corresponding bounded homomorphisms give an effective tool for studying covers in INTRODUCTION 3 free lattices. For example, with the aid of his techniques, McKenzie was able to show that deciding if a b holds in a free lattice is recursive. Chapter II gives a detailed≺ coverage of bounded homomorphisms and splitting lattices. J´onsson’s unpublished notes studied certain de- pendency relations in lattices. Although on the face it would not ap- pear so, these relations are closely related to bounded homomorphisms. These and other relations are covered thoroughly in a general setting in this chapter. The chapter includes applications of these ideas and results to several areas of lattice theory and algebra, such as projective lattices, varieties of lattices, and the equational theory of congruence lattices of algebraic systems. The dependency relations of this chapter give an efficient representation of congruences on a finite lattice, fa- cilitating fast algorithms for determining such properties as subdirect irreducibility and simplicity, which are developed in Chapter XI. In a significant paper [24], Alan Day, using his doubling construc- tion, was able to prove that the variety generated by all splitting lat- tices was in fact all lattices. This implies that free lattices are weakly atomic: every nontrivial interval contains a covering. We present two proofs of this result; the first is in Chapter II. In Chapter IV we prove a somewhat stronger version of Day’s Theorem, useful for other results on free lattices. Several basic questions about covers and about the structure of free lattices remained. Given an element, could one decide if it was covered by any other element? Could one effectively find all such covers? Is there a coverless element: one which covers no element and is covered by no element? How long could a chain of covers in a free lattice be? These questions were settled by Freese and Nation in a paper [62] which thoroughly developed the theory of covers in free lattices.
Load more

Recommended publications
  • Algebraic Aspects of Orthomodular Lattices
    Algebraic Asp ects of Orthomo dular Lattices Gunter Bruns McMaster University Hamilton Ontario LS K Canada John Harding New Mexico State University Las Cruces NM USA In this pap er we try to give an uptodate account of certain asp ects of the the ory of ortholattices abbreviated OLs orthomo dular lattices abbreviated OMLs and mo dular ortholattices abbreviated MOLs not hiding our own research in terests Since most of the questions we deal with have their origin in Universal Alge bra we start with a chapter discussing the basic concepts and results of Universal Algebra without pro ofs In the next three chapters we discuss mostly with pro ofs the basic results and standard techniques of the theory of OMLs In the remaining ve chapters we work our way to the b order of present day research mostly no or only sketchy pro ofs Chapter deals with pro ducts and sub direct pro ducts chap ter with free structures and chapter with classes of OLs dened by equations In chapter we discuss emb eddings of OLs into complete ones The last chap ter deals with questions originating in Category Theory mainly amalgamation epimorphisms and monomorphisms The later chapters of this pap er contain an abundance of op en problems We hop e that this will initiate further research Basic Universal Algebra We assume a denition of natural numb ers which makes every natural numb er n the set of all natural numb ers k n An nary op eration on a set A is a map f of n n A into A An element a A gives rise to a sequence a a an of A more commonly written with indices
    [Show full text]
  • Free Objects and Free Extensions in the Category of Frames
    FREE OBJECTS AND FREE EXTENSIONS IN THE CATEGORY OF FRAMES CONSTANTINE TSINAKIS This work is dedicated to Rick Ball on the occasion of his retirement. Abstract. This article is concerned with free objects and free extensions over posets in the category of frames. Its primary goal is to present novel representations for these objects as subdirect products of certain chains. Constructions for the corresponding objects in the category of bounded distributive lattices are also presented. 1. Discussion and the Main Results The primary aim of this note is to present novel representations for free objects and free extensions over posets in the category Frm of frames. Representations for the corresponding objects in the category D of bounded distributive lattices are also presented. Free frames are easy to describe, see for example Johnstone [5]. The free frame F over a set X is usually represented as the frame of lower sets of the free meet-semilattice with identity, or more concretely, as the the frame of lower sets of the semilattice of all finite subsets of X ordered by reverse set-inclusion. It can also be shown that F is isomorphic to the ideal completion of the free bounded distributive lattice over X. The latter representation follows from the results of the present paper, but it can also be obtained directly by using the universal properties of free distributive lattices and ideal completions of lattices. The statements for the main results will require some notation. For any poset P , we write ? ⊕ P ⊕ > for the poset obtained from P by adjoining a new least element ? and a new greatest element >.
    [Show full text]
  • Congruence Lattices of Semilattices
    PACIFIC JOURNAL OF MATHEMATICS Vol. 49, No. 1, 1973 CONGRUENCE LATTICES OF SEMILATTICES RALPH FREESE AND J. B. NATION The main result of this paper is that the class of con- gruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if 5^* is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of 5^" are groups with exponent dividing a fixed finite number. Given a variety (equational class) J^ of algebras, among the inter- esting questions we can ask about the members of SίΓ is the following: does there exist a lattice identity δ such that for each algebra A e S?~, the congruence lattice Θ(A) satisfies S? In the case that 5ίΓ has dis- tributive congruences, many strong conclusions can be drawn about the algebras of J%Γ [1, 2, 7]. In the case that 3ίΓ has permutable con- gruences or modular congruences, there is reason to hope that some similar results may be obtainable [4, 8]. A standard method of proving that a class of lattices satisfies no nontrivial lattice identities is to show that all partition lattices (lattices of equivalence relations) are contained as sublattices. The lattices of congruences of semilattices, however, are known to be pseudo-complemented [9]. It follows that the partition lattice on a three-element set (the five-element two-dimensional lattice) is not isomorphic to a sublattice of the congruence lattice of a semi- lattice, and in fact is not a homomorphic image of a sublattice of the congruence lattice of a finite semilattice.
    [Show full text]
  • G. Grätzer and J. Sichler, Free Decompositions of a Lattice
    Can. J. Math., VoL XXVII, No.2, 1975, pp. 276-285 FREE DECOMPOSITIONS OF A LATTICE G. GRATZER AND J. SICHLER 1. Introduction. Two basic questions have been raised for free products of lattices: 1, Do any two free products have a common refinement? 2. Can every lattice be decomposed into a free product of freely indecom­ posable lattices? Both questions have been around for some time and attempts at solving them were made especially after the Structure Theorem for Free Products was discovered (see G. Gratzer, H. Lasker, and C. R. Platt (3]). Partial answer to question one was supplied in A. Kostinsky (7]. In this paper we answer both questions. Our basic observation is that the proper framework for these results is the theory of free K-products, that is, free products in an arbitrary equational class K of lattices. This approach has the advantage that the answers are supplied for all equational classes of lattices. It is especially simple to answer Question 1 for equational classes having a special property (J) (see §2) describing certain sublattices of free products. We also show that many equational classes fail to have (J). 2. Results. An equational class K of lattices is called trivial if it is theclass of aU one element lattices; otherwise it is nontrivial. THEOREM 1, Let K be a nontrivial equational class of lattices. For any L in K, any two representations of L as a free K-product have a common refinement. It is easy to state what the common refinement is. To simplify our notation, we agree that we use the "internal" definition of free K-product, that is, the free K-factors are considered as sublattices of the free K-products.
    [Show full text]
  • Lecture Notes in Mathematics 2029
    Lecture Notes in Mathematics 2029 Editors: J.-M. Morel, Cachan B. Teissier, Paris For further volumes: http://www.springer.com/series/304 • Pierre Gillibert • Friedrich Wehrung From Objects to Diagrams for Ranges of Functors 123 Pierre Gillibert Friedrich Wehrung Charles University in Prague University of Caen, LMNO, CNRS Department of Mathematics UMR 6139, Sokolovsk´a83 Department of Mathematics 18600 Prague Praha 14032 Caen, Cedex Czech Republic France [email protected] [email protected] ISBN 978-3-642-21773-9 e-ISBN 978-3-642-21774-6 DOI 10.1007/978-3-642-21774-6 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011933135 Mathematics Subject Classification (2010): Primary 18A30, 18A25, 18A20, 18A35; Secondary 03E05, 05D10, 06A07, 06A12 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
    [Show full text]
  • Congruence Lattices of Free Lattices in Non-Distributive Varieties Miroslav Ploscica, Jiri Tuma, Friedrich Wehrung
    Congruence lattices of free lattices in non-distributive varieties Miroslav Ploscica, Jiri Tuma, Friedrich Wehrung To cite this version: Miroslav Ploscica, Jiri Tuma, Friedrich Wehrung. Congruence lattices of free lattices in non- distributive varieties. Colloquium Mathematicum, 1998, 76, no. 2, pp.269-278. hal-00004064 HAL Id: hal-00004064 https://hal.archives-ouvertes.fr/hal-00004064 Submitted on 25 Jan 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CONGRUENCE LATTICES OF FREE LATTICES IN NON-DISTRIBUTIVE VARIETIES MIROSLAV PLOSˇCICA,ˇ JIRˇ´I TUMA,˚ AND FRIEDRICH WEHRUNG Abstract. We prove that for any free lattice F with at least ℵ2 generators in any non-distributive variety of lattices, there exists no sectionally complemen- ted lattice L with congruence lattice isomorphic to the one of F . This solves a question formulated by Gr¨atzer and Schmidt in 1962. This yields in turn further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring. Introduction One of the oldest and most famous open problems in lattice theory, the Congru- ence Lattice Problem, is to decide whether for every distributive (join-) semilattice S with zero, there exists a lattice L such that the semilattice C(L) of compact congruences of L (the congruence semilattice of L) is isomorphic to S.
    [Show full text]
  • Arxiv:1901.07638V3 [Math.GR] 8 Jan 2020 Rpitsbitdt Arxiv to Submitted Preprint E Gi 4.I Hspprw Show, We Paper This in [4]
    Orders on groups, and spectral spaces of lattice-groups Almudena Colacitoa, Vincenzo Marrab aMathematisches Institut, Universitat¨ Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland bDipartimento di Matematica “Federigo Enriques”, Universita` degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy Abstract Extending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups. Keywords: Partially ordered group, Total order, Right order, Free lattice-ordered group, Spectral space, Stone duality. 2010 MSC: Primary: 06F15. Secondary: 06E15, 03C05, 08B15. 1. Introduction A right order on a group G is a total order 6 on G such that x 6 y implies xt 6 yt, for all x,y,t ∈ G. There are at least two distinct motivations for studying such orders on groups. First, a countable group admits a right order if, and only if, it acts faithfully on the real line by orientation-preserving homeomorphisms. This, as far as we know, is folklore; see [11, Theorem 6.8] for a proof. (While we use right orders in this paper, other authors prefer left orders as in [11], the difference being immaterial.) The result indicates that orders on groups play a roleˆ in topological dynamics. For more on this, besides Section 6.5 of Ghys’ beautiful survey [11], see the research monograph [4].
    [Show full text]
  • Free Lattices, 1995 41 Hal L
    http://dx.doi.org/10.1090/surv/042 Recent Titles in This Series 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1993 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°9 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E.
    [Show full text]
  • Free-Lattice Functors Weakly Preserve Epi-Pullbacks
    Free-lattice functors weakly preserve epi-pullbacks H. Peter Gumm, Ralph Freese March 18, 2021 Abstract Suppose p(x,y,z) and q(x,y,z) are terms. If there is a common “an- cestor” term s(z1,z2,z3,z4) specializing to p and q through identifying some variables p(x,y,z) ≈ s(x,y,z,z) q(x,y,z) ≈ s(x,x,y,z), then the equation p(x,x,z) ≈ q(x,z,z) is trivially obtained by syntactic unification of s(x,y,z,z) with s(x,x,y,z). In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, the converse is true, too. Given terms p,q, and an equation p(u1,...,um) ≈ q(v1,...,vn) (1) where {u1,...,um} = {v1,...,vn}, there is always an “ancestor term” s(z1,...,zr) such that p(x1,...,xm) and q(y1,...,yn) arise as substitu- tion instances of s, whose unification results in the original equation (1). In category theoretic terms the above proposition, when restricted to lattices, has a much more concise formulation: Free-lattice functors weakly preserves pullbacks of epis. arXiv:2103.09566v1 [math.RA] 17 Mar 2021 1 Introduction The motivation of this study arose from coalgebra. When studying F -coalgebras for a type functor F , the limit preservation properties of F are known to exert a crucial influence on the structure theory of the class of all F -coalgebras. In particular, weak preservation of pullbacks is a property which many familiar Set- endofunctors enjoy.
    [Show full text]
  • The Completion of a Lattice Ordered Group
    THE COMPLETION OF A LATTICE ORDERED GROUP , PAUL CONRAD and DONALD McALISTER 1 (Received 8 May 1967) 1. Introduction A lattice ordered group ('/-group') is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete /-group is archimedean and hence abelian, and each archimedean /-group has a completion in the sense of the following theorem. THEOREM 1.1. // G is an archimedean l-group, then there exists a complete l-group GA with the following properties (1) G is an l-subgroup of G*, and (2) if heGA, then h = y{geG\g ^ h). Moreover, if H is any complete l-group with these properties, then there exists a unique l-isomorphism a of G* onto H such that go = g for all g eG. Note that for each h e C\ —h = \/{g e G\g ^ —h} so that (2') h = /\{geG\g^h} also holds. The complete /-group G* described in this theorem is the Dedekind- MacNeille 'completion by cuts' of G (see [4] or [12] for details of its con- struction). Although Theorem 1.1 gives a more abstract characterisation of G* it does not enable one to compute the completion of an archimedean /-group. In Theorem 2.4 we give an abstract characterisation of GA which makes no mention of infinite suprema or infima, and using this result we can precisely describe the completion of an archimedean /-group with a basis. As a further application of Theorem 2.4 we show that the completion of an /-group of real valued functions is an /-group of real valued functions over the same domain (Theorem 3.2) and give a reasonably decent description of this completion (Theorem 3.3).
    [Show full text]
  • Arxiv:Math/0501375V1
    A SURVEY OF RECENT RESULTS ON CONGRUENCE LATTICES OF LATTICES JIRˇ´I TUMA˚ AND FRIEDRICH WEHRUNG Dedicated to Ralph McKenzie on his 60-th birthday Abstract. We review recent results on congruence lattices of (infinite) lat- tices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results. Contents 1. Introduction 1 2. Uniform Refinement Properties 3 3. The M3hLi construction, tensor product, and box product 9 4. The functor Conc on partial lattices 13 5. Lifting diagrams of semilattices by diagrams of partial lattices 14 6. Extensions of partial lattices to lattices 17 7. Connections to ring theory 20 8. Dual topological spaces 21 9. Open problems 23 Acknowledgments 25 Added in proof 25 References 25 1. Introduction For a lattice L, the congruence lattice of L, denoted here by Con L, is the lattice of all congruences of L under inclusion. As the congruence lattice of any algebraic arXiv:math/0501375v1 [math.GM] 22 Jan 2005 system, the lattice Con L is algebraic. The compact elements of Con L are the finitely generated congruences, that is, the congruences of the form ΘL(ai,bi), _ i<n Date: September 21, 2018. 2000 Mathematics Subject Classification. 06B10, 06E05. Key words and phrases. Lattice, congruence, box product, partial lattice, amalgamation, reg- ular ring, locally matricial ring, dual topological space. The first author was partially supported by GA UK grant no. 162/1999 and by GA CR grant no. 201/99. The second author was partially supported by the Fund of Mobility of the Charles University (Prague), by FRVS grant no.
    [Show full text]
  • Properties of Lattices of Subquasivarieties
    APPENDIX A Properties of Lattices of Subquasivarieties A.1. Representations of Quasivariety Lattices The span of nearly two decades since the publication of Viktor Gor- bunov’s book on quasivarieties [77] has made it desirable to have an updated review of what is known about lattices of quasivarieties. This appendix pro- vides a summary of their basic properties, with references to the literature. A.2. Representations of Quasivariety Lattices Representations of a lattice can reveal structural properties, so let us be- gin there. For a quasivariety K,letLq(K) denote the lattice of quasivarieties contained in K, ordered by inclusion. No assumption is made yet on the type of the structures in K, but the type will generally have both operations and relations. Of course Lq(K) is a complete lattice. The theory of a quasivariety K is the set of all quasi-equations holding in K. The theory of a subquasivariety Q ≤ K contains the theory of K. (More quasi-equations means fewer models.) The lattice QTh(K) of all quasi- equational theories extending the theory of K, ordered by containment, is dually isomorphic to the lattice of subquasivarieties Lq(K). In Adaricheva and Nation [19], it is shown that QTh(K)canberepre- sented as the congruence lattice of a semilattice with operators. That is, ∼ there is an algebra S such that QTh(K) = Con S,whereS =(S, ∨, 0,F)isa join semilattice with 0 endowed with a set of operations f : S → S such that f(x ∨ y)=f(x) ∨ f(y)andf(0) = 0 for all f ∈ F .
    [Show full text]