Free Lattices

If x, y and z are elements of a lattice, then x ∨ (y ∨ (x ∧ z)) = x ∨ y is always true, while x ∨ y = z is usually not true. Is there an algorithm that, given two lattice expressions p and q, determines whether p = q holds for every substitution of the variables in every lattice? The answer is yes, and finding this algorithm (Corollary to Theorem 6.2) is our original motivation for studying free lattices. We say that a lattice L is generated by a set X ⊆ L if no proper sublattice of L contains X. In terms of the subalgebra closure operator Sg introduced in Chapter 3, this means Sg(X) = L.

[1]  Alan Day,et al.  A Simple Solution to the Word Problem for Lattices , 1970, Canadian Mathematical Bulletin.

[2]  Jean-Pierre Jouannaud,et al.  Open Problems in Rewriting , 1991, RTA.

[3]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[4]  Ivan Rival,et al.  A note on Whitman's property for free lattices , 1974 .

[5]  James B. Nation,et al.  A REPORT ON SUBLATTICES OF A FREE LATTICE , 1977 .

[6]  R. McKenzie.  Equational bases and nonmodular lattice varieties , 1972 .

[7]  G. Grätzer.  General Lattice Theory , 1978 .

[8]  George Markowsky,et al.  The factorization and representation of lattices , 1975 .

[9]  Ivan Rival,et al.  Planar Sublattices of a Free Lattice. I , 1978, Canadian Journal of Mathematics.

[10]  N. Kehayopulu.  ON ORDERED Γ-SEMIGROUPS , 2010 .

[11]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[12]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[13]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[14]  Richard A. Dean,et al.  Component subsets of the free lattice on $n$ generators , 1956 .

[15]  R. McKenzie,et al.  Commutator theory for congruence modular varieties , 1989 .

[16]  R. Wille.  ON LATTICES FREELY GENERATED BY FINITE PARTIALLY ORDERED SETS , 1977 .

[17]  Trevor Evans.  On multiplicative systems defined by generators and relations , 1953 .

[18]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[19]  R. Freese.  Directions in lattice theory , 1994 .

[20]  Donald E. Knuth,et al.  Simple Word Problems in Universal Algebras††The work reported in this paper was supported in part by the U.S. Office of Naval Research. , 1970 .

[21]  Alan Day,et al.  Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices , 1979, Canadian Journal of Mathematics.

[22]  Daniel J. Kleitman,et al.  Maximal sized antichains in partial orders , 1971, Discret. Math..

[23]  R. A. Dean,et al.  Completely free lattices generated by partially ordered sets , 1956 .

[24]  Deko V. Dekov,et al.  Embeddability and the word problem , 1995, Journal of Symbolic Logic.

[25]  Identities in congruence lattices of universal algebras , 1977 .

[26]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[27]  James B. Nation,et al.  CONGRUENCE LATTICES OF SEMILATTICES , 1973 .

[28]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[29]  Alan Day,et al.  Congruence normality: The characterization of the doubling class of convex sets , 1994 .

[30]  R. Freese.  Connected components of the covering relation in free lattices , 1985 .

[31]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[32]  A. Davis,et al.  A characterization of complete lattices , 1955 .

[33]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[34]  Lattice varieties covering V(L1) , 1986 .

[35]  Ralph Freese.  Free lattice algorithms , 1987 .

[36]  Vojtech Rödl,et al.  Fast Recognition of Rings and Lattices , 1981, FCT.

[37]  A. B. Kahn,et al.  Topological sorting of large networks , 1962, CACM.

[38]  R. P. Dilworth.  The Structure of Relatively Complemented Lattices , 1950 .

[39]  Keith A. Kearnes,et al.  Congruence Lattices of Congruence Semidistributive Algebras , 1997 .

[40]  Ralph Freese,et al.  The variety of modular lattices is not generated by its finite members , 1979 .

[41]  Finitely Presented Lattices: Continuity and Semidistributivity , 1990 .

[42]  Ralph McKenzie.  Equational Bases for Lattice Theories. , 1970 .

[43]  Rudolf Wille.  Eine Charakterisierung endlicher, ordnungspolynomvollständiger Verbände , 1977 .

[44]  Almost distributive lattice varieties , 1985 .

[45]  Ralph Freese.  Finitely presented lattices: canonical forms and the covering relation , 1989 .

[46]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[47]  R. P. Dilworth.  Some Combinatorial Problems on Partially Ordered Sets , 1990 .

[48]  A. Kostinsky,et al.  Projective lattices and bounded homomorphisms. , 1972 .

[49]  Bjarni Jónsson,et al.  Sublattices of a Free Lattice , 1961, Canadian Journal of Mathematics.

[50]  Trevor Evans,et al.  The Word Problem for Abstract Algebras , 1951 .

[51]  E. Szpilrajn.  Sur l'extension de l'ordre partiel , 1930 .

[52]  J. Schmid,et al.  Drawing orders using less ink , 1992 .

[53]  Bjarni Jónsson,et al.  Finite sublattices of a free lattice , 1982 .

[54]  J. B. Nation,et al.  Projective Lattices , 1995 .

[55]  W. Marsden.  I and J , 2012 .

[56]  On lower bounded lattices , 2001 .

[57]  Free lattices over join-trivial partial lattices , 1990 .

[58]  Rudolf Wille,et al.  Lattices freely generated by partially ordered sets: which can be "drawn"? , 1979 .

[59]  Klaus Simon,et al.  An Improved Algorithm for Transitive Closure on Acyclic Digraphs , 1986, Theor. Comput. Sci..

[60]  Jean Paul Bordat.  Efficient polynomial algorithms for distributive lattices , 1991, Discret. Appl. Math..

[61]  Ralph FREESE.  An application of dilworth's lattice of maximal antichains , 1974, Discret. Math..

[62]  L. Fuchs.  Partially Ordered Algebraic Systems , 2011 .

[63]  R. Dean.  Free Lattices Generated by Partially Ordered Sets and Preserving Bounds , 1964, Canadian Journal of Mathematics.

[64]  J. Kung,et al.  The Impact of the Chain Decomposition Theorem on Classical Combinatorics , 1990 .

[65]  Congruence Normal Covers of Finitely Generated Lattice Varieties , 1992, Canadian Mathematical Bulletin.

[66]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[67]  James B. Nation,et al.  Covers in free lattices , 1985 .

[68]  J. C. C. McKinsey,et al.  The decision problem for some classes of sentences without quantifiers , 1943, Journal of Symbolic Logic.

[69]  Carlo Batini,et al.  Automatic graph drawing and readability of diagrams , 1988, IEEE Trans. Syst. Man Cybern..

[70]  The order dimension of relatively free lattices , 1990 .

[71]  Some varieties of semidistributive lattices , 1985 .

[72]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

[73]  Alan Day.  Splitting lattices generate all lattices , 1977 .

[74]  James B. Nation,et al.  Term Rewrite Systems for Lattice Theory , 1993, J. Symb. Comput..

[75]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[76]  Alan Day,et al.  A Characterization of Identities Implying Congruence Modularity I , 1980, Canadian Journal of Mathematics.

[77]  B. Jonnson.  Algebras Whose Congruence Lattices are Distributive. , 1967 .

[78]  John Beidler,et al.  Data Structures and Algorithms , 1996, Wiley Encyclopedia of Computer Science and Engineering.

[79]  Harry B. Hunt,et al.  On the Computational Complexity of Algebra on Lattices , 1987, SIAM J. Comput..

[80]  J. B. Nation.  On partially ordered sets embeddable in a free lattice , 1984 .

[81]  Alan Day,et al.  Doubling Constructions in Lattice Theory , 1992, Canadian Journal of Mathematics.

[82]  Kenneth P. Bogart,et al.  Introductory Combinatorics , 1977 .

[83]  Reflections on lower bounded lattices , 2005 .

[84]  Ivan Rival,et al.  Lattice varieties covering the smallest nonmodular variety. , 1979 .

[85]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .

[86]  Stanley Burris,et al.  Term Rewrite Rules for Finite Fields , 1991, Int. J. Algebra Comput..

[87]  David Magagnosc.  Cuts and decompositions: structure and algorithms , 1987 .

[88]  Richard A. Dean,et al.  FREE LATTICES WITH INFINITE OPERATIONS , 1959 .

[89]  Steven T. Tschantz.  Infinite intervals in free lattices , 1990 .

[90]  Václav Koubek,et al.  A Reduct-and-Closure Algorithm for Graphs , 1979, MFCS.

[91]  B. Jónsson,et al.  Distributive Sublattices of a Free Lattice , 1961, Canadian Journal of Mathematics - Journal Canadien de Mathematiques.

[92]  J. Berman.  Sublattices of intervals in relatively free lattices , 1972 .

[93]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[94]  D. Papert,et al.  Congruence relations in semilattices , 1964 .

[95]  James B. Nation,et al.  An approach to lattice varieties of finite height , 1990 .

[96]  Václav Koubek,et al.  Testing of Properties of Finite Algebras , 1980, ICALP.

[97]  Václav Koubek,et al.  Fast Algorithms Constructing Minimal Subalgebras, Congruences, and Ideals in a Finite Algebra , 1985, Theor. Comput. Sci..

[98]  K. A. Baker,et al.  Finite equational bases for finite algebras in a congruence-distributive equational class* , 1977 .

[99]  G. Birkhoff,et al.  On the Structure of Abstract Algebras , 1935 .

[100]  R. Freese.  Some Order Theoretic Questions About Free Lattices and Free Modular Lattices , 1982 .

[101]  Ralph Freese,et al.  Free modular lattices , 1980 .

[102]  H. L. Rolf,et al.  THE FREE LATTICE GENERATED BY A SET OF CHAINS , 1958 .