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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The factorization and representation of lattices
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by George Markowsky PDF
Trans. Amer. Math. Soc. 203 (1975), 185-200 Request permission

Abstract:

For a complete lattice $L$, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say $L$ is a jm-lattice) we define the poset of irreducibles $P(L)$ to be the poset (of height one) $J(L) \cup M(L)(J(L)$ is the set of completely join-irreducibles and $M(L)$ is the set of completely meet-irreducibles) ordered as follows: $a{ < _{P(L)}}b$ if and only if $a \in J(L),b \in M(L)$, and $a \nleqslant { _L}b$. For a jm-lattice $L$, the automorphism groups of $L$ and $P(L)$ are isomorphic, $L$ can be reconstructed from $P(L)$, and the irreducible factorization of $L$ can be gotten from the components of $P(L)$. 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 $P(L)$). Thus $P(L)$ 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 $P(L)$ for some jm-lattice $L$. We also characterize those posets of height 1 which are $P(L)$ for a completely distributive jm-lattice, as well as those posets which are $P(L)$ for some geometric lattice $L$. More generally, if $L$ is a complete lattice, many of the above arguments apply if we use “join-spanning” and “meet-spanning” subsets of $L$, instead of $J(L)$ and $M(L)$. If $L$ is an arbitrary lattice, the same arguments apply to “join-generating” and “meet-generating” subsets of $L$.
References
  • Garrett Birkhoff, Lattice theory, 3rd ed., 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.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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