Covers in free lattices
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 by Ralph Freese and J. B. Nation PDF
 Trans. Amer. Math. Soc. 288 (1985), 142 Request permission
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 threeelement chains.References

P. Crawley and R. P. Dilworth, Algebraic theory of lattices, PrenticeHall, Englewood Cliffs, N. J., 1973.
 Alan Day, A simple solution to the word problem for lattices, Canad. Math. Bull. 13 (1970), 253–254. MR 268092, DOI 10.4153/CMB19700510
 Alan Day, Splitting lattices generate all lattices, Algebra Universalis 7 (1977), no. 2, 163–169. MR 434897, DOI 10.1007/BF02485425
 Alan Day, Characterizations of finite lattices that are boundedhomomorphic images of sublattices of free lattices, Canadian J. Math. 31 (1979), no. 1, 69–78. MR 518707, DOI 10.4153/CJM1979008x
 Alan Day and J. B. Nation, A note on finite sublattices of free lattices, Algebra Universalis 15 (1982), no. 1, 90–94. MR 663955, DOI 10.1007/BF02483711
 Richard A. Dean, Coverings in free lattices, Bull. Amer. Math. Soc. 67 (1961), 548–549. MR 133264, DOI 10.1090/S000299041961106812
 Ralph Freese, Ideal lattices of lattices, Pacific J. Math. 57 (1975), no. 1, 125–133. MR 371751 —, Some order theoretic questions about free lattices and free modular lattices, Proc. Banff Sympos. on Ordered Sets, Reidel, Dordrecht, 1982.
 Ralph Freese and J. B. Nation, Projective lattices, Pacific J. Math. 75 (1978), no. 1, 93–106. MR 500031 G. Grätzer, General lattice theory, Academic Press, New York, 1978.
 Bjarni Jónsson, Sublattices of a free lattice, Canadian J. Math. 13 (1961), 256–264. MR 123493, DOI 10.4153/CJM19610210
 Bjarni Jónsson and James E. Kiefer, Finite sublattices of a free lattice, Canadian J. Math. 14 (1962), 487–497. MR 137667, DOI 10.4153/CJM19620401
 B. Jónsson and J. B. Nation, A report on sublattices of a free lattice, Contributions to universal algebra (Colloq., József Attila Univ., Szeged, 1975) Colloq. Math. Soc. János Bolyai, Vol. 17, NorthHolland, Amsterdam, 1977, pp. 223–257. MR 0472614
 Ralph McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1–43. MR 313141, DOI 10.1090/S00029947197203131411
 J. B. Nation, Bounded finite lattices, Universal algebra (Esztergom, 1977) Colloq. Math. Soc. János Bolyai, vol. 29, NorthHolland, AmsterdamNew York, 1982, pp. 531–533. MR 660892
 J. B. Nation, Finite sublattices of a free lattice, Trans. Amer. Math. Soc. 269 (1982), no. 1, 311–337. MR 637041, DOI 10.1090/S00029947198206370419
 Philip M. Whitman, Free lattices, Ann. of Math. (2) 42 (1941), 325–330. MR 3614, DOI 10.2307/1969001
 Philip M. Whitman, Free lattices. II, Ann. of Math. (2) 43 (1942), 104–115. MR 6143, DOI 10.2307/1968883
 Bjarni Jónsson, Varieties of lattices: some open problems, Universal algebra (Esztergom, 1977) Colloq. Math. Soc. János Bolyai, vol. 29, NorthHolland, AmsterdamNew York, 1982, pp. 421–436. MR 660878
Additional Information
 © Copyright 1985 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 288 (1985), 142
 MSC: Primary 06B25
 DOI: https://doi.org/10.1090/S00029947198507730444
 MathSciNet review: 773044