Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Covers in free lattices


Authors: Ralph Freese and J. B. Nation
Journal: Trans. Amer. Math. Soc. 288 (1985), 1-42
MSC: Primary 06B25
DOI: https://doi.org/10.1090/S0002-9947-1985-0773044-4
MathSciNet review: 773044
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the covering relation $ (u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $ w \in {\text{FL}}(X)$, finds all the elements which cover or are covered by $ w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in $ {\text{FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.


References [Enhancements On Off] (What's this?)

  • [1] P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, N. J., 1973.
  • [2] A. Day, A simple solution to the word problem for lattices, Canad. Math. Bull. 13 (1970), 253-254. MR 0268092 (42:2991)
  • [3] -, Splitting lattices generate all lattices, Algebra Universalis 7 (1977), 163-169. MR 0434897 (55:7861)
  • [4] -, Characterizations of lattices that are bounded-homomorphic images of sublattices of free lattices, Canad. J. Math. 31 (1979), 69-78. MR 518707 (81h:06004)
  • [5] A. Day and J. B. Nation, A note on finite sublattices of free lattices, Algebra Universalis 15 (1982), 90-94. MR 663955 (83k:06011)
  • [6] R. A. Dean, Coverings in free lattices, Bull. Amer. Math. Soc. 67 (1961), 548-549. MR 0133264 (24:A3098)
  • [7] R. Freese, Ideal lattices of lattices, Pacific J. Math. 57 (1975), 125-133. MR 0371751 (51:7968)
  • [8] -, Some order theoretic questions about free lattices and free modular lattices, Proc. Banff Sympos. on Ordered Sets, Reidel, Dordrecht, 1982.
  • [9] R. Freese and J. B. Nation, Projective lattices, Pacific J. Math. 75 (1978), 93-106. MR 500031 (80c:06012)
  • [10] G. Grätzer, General lattice theory, Academic Press, New York, 1978.
  • [11] B. Jónsson, Sublattices of a free lattice, Canad. J. Math. 13 (1961), 256-264. MR 0123493 (23:A818)
  • [12] B. Jónsson and J. Kiefer, Finite sublattices of a free lattice, Canad. J. Math. 14 (1962), 487-497. MR 0137667 (25:1117)
  • [13] B. Jónsson and J. B. Nation, A report on sublattices of a free lattice, Colloq. Math. Soc. János Bolyai, Contributions to Universal Algebra (Szeged), Vol. 17, North-Holland, Amsterdam, 1977, pp. 223-257. MR 0472614 (57:12310)
  • [14] R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43. MR 0313141 (47:1696)
  • [15] J. B. Nation, Bounded finite lattices, Colloq. Math. Soc. János Bolyai, Contributions to Universal Algebra (Esztergom), Vol. 29, North-Holland, Amsterdam, 1982, pp. 531-533. MR 660892 (83f:06009)
  • [16] -, Finite sublattices of a free lattice, Trans. Amer. Math. Soc. 269 (1982), 311-337. MR 637041 (83b:06008)
  • [17] P. Whitman, Free lattices, Ann. of Math. (2) 42 (1941), 325-330. MR 0003614 (2:244f)
  • [18] -, Free lattices. II, Ann. of Math. (2) 43 (1942), 104-115. MR 0006143 (3:261d)
  • [19] B. Jónsson, Varieties of lattices: Some open problems, Colloq. Math. Soc. János Bolyai, Contributions to Universal Algebra (Esztergom), Vol. 29, North-Holland, Amsterdam, 1982, pp. 421-436. MR 660878 (83g:06010)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06B25

Retrieve articles in all journals with MSC: 06B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0773044-4
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society