The factorization and representation of lattices
Author: George Markowsky
Journal: Trans. Amer. Math. Soc. 203 (1975), 185-200
MSC: Primary 06A20
MathSciNet review: 0360386
Full-text PDF Free Access
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 .
-  Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
-  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
-  P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, N. J., 1973.
-  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
-  -, Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices, Ph. D. Thesis, Harvard University, Cambridge, Mass., June 1973.
- G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638. MR 0227053 (37:2638)
- H. H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: II. Combinatorial geometries, (preliminary edition), M. I. T. Press, Cambridge, Mass., 1970. MR 0290980 (45:74)
- P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, N. J., 1973.
- G. Markowsky, Some combinatorial aspects of lattice theory, Proc. Houston Lattice Theory Conf. , Univ. of Houston Press, 1973, pp. 36-38. MR 0396352 (53:219)
- -, Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices, Ph. D. Thesis, Harvard University, Cambridge, Mass., June 1973.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 06A20
Retrieve articles in all journals with MSC: 06A20
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