The factorization and representation of lattices

Author:
George Markowsky

Journal:
Trans. Amer. Math. Soc. **203** (1975), 185-200

MSC:
Primary 06A20

DOI:
https://doi.org/10.1090/S0002-9947-1975-0360386-3

MathSciNet review:
0360386

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Abstract: For a complete lattice , in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say is a *jm-lattice*) we define the *poset of irreducibles * to be the poset (of height one) is the set of completely join-irreducibles and is the set of completely meet-irreducibles) ordered as follows: if and only if , and . For a jm-lattice , the automorphism groups of and are isomorphic, can be reconstructed from , and the irreducible factorization of can be gotten from the components of . In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of ). Thus extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices.

We characterize those posets of height 1 which are for some jm-lattice . We also characterize those posets of height 1 which are for a completely distributive jm-lattice, as well as those posets which are for some geometric lattice .

More generally, if is a complete lattice, many of the above arguments apply if we use ``join-spanning'' and ``meet-spanning'' subsets of , instead of and . If is an arbitrary lattice, the same arguments apply to ``join-generating'' and ``meet-generating'' subsets of .

**[1]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[2]**Henry H. Crapo and Gian-Carlo Rota,*On the foundations of combinatorial theory: Combinatorial geometries*, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR**0290980****[3]**P. Crawley and R. P. Dilworth,*Algebraic theory of lattices*, Prentice-Hall, Englewood Cliffs, N. J., 1973.**[4]**George Markowsky,*Some combinatorial aspects of lattice theory*, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973) Dept. Math., Univ. Houston, Houston, Tex., 1973, pp. 36–68. MR**0396352****[5]**-,*Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices*, Ph. D. Thesis, Harvard University, Cambridge, Mass., June 1973.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0360386-3

Keywords:
Poset of irreducibles,
completely join-irreducible,
Galois connection,
irreducible factorization,
representations,
group of automorphism,
geometric lattice,
poset of join-irreducibles,
join-spanning set,
distributive lattice,
separators,
center

Article copyright:
© Copyright 1975
American Mathematical Society