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Primes, irreducibles and extremal lattices


This paper studies certain types of join and meet-irreducibles called coprimes and primes. These elements can be used to characterize certain types of lattices. For example, a lattice is distributive if and only if every join-irreducible is coprime. Similarly, a lattice is meet-pseudocomplemented if and only if each atom is coprime. Furthermore, these elements naturally decompose lattices into sublattices so that often properties of the original lattice can be deduced from properties of the sublattice. Not every lattice has primes and coprimes. This paper shows that lattices which are “long enough” must have primes and coprimes and that these elements and the resulting decompositions can be used to study such lattices.

The length of every finite lattice is bounded above by the minimum of the number of meet-irreducibles (meet-rank) and the number of join-irreducibles (join-rank) that it has. This paper studies lattices for which length=join-rank or length=meet-rank. These are called p-extremal lattices and they have interesting decompositions and properties. For example, ranked, p-extremal lattices are either lower locally distributive (join-rank=length), upper locally distributive (meet-rank=length) or distributive (join-rank=meet-rank=length). In the absence of the Jordan-Dedekind chain condition, p-extremal lattices still have many interesting properties. Of special interest are the lattices that satisfy both equalities. Such lattices are called extremal; this class includes distributive lattices and the associativity lattices of Tamari. Even though they have interesting decompositions, extremal lattices cannot be characterized algebraically since any finite lattice can be embedded as a subinterval into an extremal lattice. This paper shows how prime and coprime elements, and the poset of irreducibles can be used to analyze p-extremal and other types of lattices.

The results presented in this paper are used to deduce many key properties of the Tamari lattices. These lattices behave much like distributive lattices even though they violate the Jordan-Dedekind chain condition very strongly having maximal chains that vary in length from N-1 to N(N-1)/2 where N is a parameter used in the construction of these lattices.

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  1. 1.

    S. P.Avann (1961) Application of the join-irreducible excess function to semi-modular lattices, Math. Annalen 142, 345–354.

    Google Scholar 

  2. 2.

    V. K.Balachandran (1955) On complete lattices and a problem of Birkhoff and Frink, Proc. AMS 6, 548–553.

    Google Scholar 

  3. 3.

    B.Banaschewski (1985) Prime elements from prime ideals, Order 2, 211–213.

    Google Scholar 

  4. 4.

    M. K. Bennett and Garrett Birkhoff (1990) Two families of Newman lattices, preprint.

  5. 5.

    Garrett, Birkhoff 1940 Lattice Theory, AMS Colloquium Publ., Vol. 25, 1st ed., Providence.

  6. 6.

    Garrett, Birkhoff 1967 Lattice Theory. AMS Colloquium Publ., Vol. 25, 3rd ed., Providence.

  7. 7.

    PeterCrawley and Robert P.Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  8. 8.

    C.Chameni-Nembua and B.Monjardet (1992) Les trellis pseudocomplémentés finis, European J. Combinatorics 13, 89–107. An English summary of these results is to appear in Discrete Mathematics.

    Google Scholar 

  9. 9.

    Paul H.Edelman and Michael E.Saks (1988) Combinatorial representation and convex dimension of convex geometries, Order 5, 23–32.

    Google Scholar 

  10. 10.

    MarcelErné (1988) Weak distributive laws and their role in lattices of congruences and equational theories, Algebra Universalis 25, 290–321.

    Google Scholar 

  11. 11.

    H. S.Gaskill and J. B.Nation (1981) Join-prime elements in semidistributive lattices, Algebra Universalis 12, 352–359.

    Google Scholar 

  12. 12.

    E.Graczynska (1977) On the sums of double systems of lattices and DS-congruences of lattices, Coll. Math. Soc. Janos. Bolyai 17, 161–166.

    Google Scholar 

  13. 13.

    GeorgeGrätzer (1978) General Lattice Theory, Academic Press, NY.

    Google Scholar 

  14. 14.

    CurtisGreene and GeorgeMarkowsky (1974) A combinatiorial test for local distributivity, Research Report RC5129, IBM T.J. Watson Research Center, Yorktown Heights, NY. The proof of the main result also appears in Markowsky 1980.

    Google Scholar 

  15. 15.

    S.Huang and D.Tamari (1972) A simple proof for the lattice property, J. Combinatorial Theory A13, 7–13.

    Google Scholar 

  16. 16.

    B.Korte, L.Lovasz, and R.Schrader (1991) Greedoids, Algorithms and Combinatorics 4, Springer Verlag, Berlin.

    Google Scholar 

  17. 17.

    George Markowsky (1973) Some combinatorial aspects of lattice theory with applications of the enumeration of free distributive lattices, Ph.D. Thesis, Harvard University.

  18. 18.

    George Markowsky (1973) Some Combinatorial Aspects of Lattice Theory, Proc. Univ. of Houston Lattice Theory Conf., Houston 1973, 36–68.

  19. 19.

    GeorgeMarkowsky (1975) The factorization and representation of lattices, Transactions of the American Mathematical Society 203, 185–200.

    Google Scholar 

  20. 20.

    GeorgeMarkowsky (1980) The representation of posets and lattices by sets, Algebra Universalis 11, 173–192.

    Google Scholar 

  21. 21.

    GeorgeMarkowsky and MarioPetrich (1977) Subprojective lattices and projective geometry, Journal of Algebra 48(2), 305–320.

    Article  Google Scholar 

  22. 22.

    Bernard Monjardet (1990) The consequences of Dilworth's work on lattices with unique irreducible decompositions, pp. 192–199 & 201 in The Dilworth Theorems, ed. by K.P. Bogart, R. Freese, and J. P. S. Kung, Birkhauser.

  23. 23.

    IvanRival (1982) The retract construction, pp. 97–122 in Ordered Sets, ed. by IvanRival, D. Reidel, Dordrecht.

    Google Scholar 

  24. 24.

    Richard P.Stanley, (1986) Enumerative Combinatorics, Vol. I. Wadsworth, Monterey CA.

    Google Scholar 

  25. 25.

    GaborSzasz (1963) Introduction to Lattice Theory. Academic Press, New York.

    Google Scholar 

  26. 26.

    AlasdairUrquhart (1978) A topological representation theory for lattices, Algebra Universalis 8, 45–58.

    Google Scholar 

  27. 27.

    RudolfWille (1982) Restructuring lattice theory: an approach based on hierarchies of concepts, pp. 445–470 in Ordered Sets, ed. by IvanRival, D. Reidel, Dordrecht.

    Google Scholar 

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Communicated by I. Rival

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Markowsky, G. Primes, irreducibles and extremal lattices. Order 9, 265–290 (1992).

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Mathematics Subject Classification (1991)

  • 06B05

Key words

  • Join-irreducible
  • meet-irreducible
  • prime
  • coprime
  • extremal
  • distributive
  • locally distributive
  • Jordan-Dedekind chain condition
  • retract
  • poset of irreducibles
  • context
  • Tamari lattices