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논문 기본 정보
- 자료유형
- 학술저널
- 저자정보
- 발행연도
- 2020.1
- 수록면
- 1 - 20 (20page)
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A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice P can be constructed from P by removing a particular family IL of its irreducible intervals. Applying this in the case that P is a product of a finite set C of chains, we get a one-to-one correspondence L → D_P(L) between the sublattices of P and the preorders spanned by a canonical sublattice C of P. We then show that L is a tight sublattice of the product of chains P if and only if D_P(L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of P and the posets spanned by its poset J(P) of non-zero join-irreducible elements. With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice L into products of chains, and the chain decompositions of its poset J(L) of non-zero join-irreducible elements.
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참고문헌 (9)
참고문헌 신청
G. Birkhoff / 1937 / Rings of sets / Duke Math. J. 3(3):443~454
R. Dilworth / 1950 / A decomposition theorem for partially ordered sets / Ann. of Math 51:161~166
R. Dilworth / 1960 / Some combinatorial problems on partially ordered sets / Proc. Sympos. Appl. Math 10:85~90
G. Gratzer / 2011 / Lattice theory: foundation / Birkhauser/Springer Basel
K. M. Koh / 1983 / On the lattice of maximum-sized antichains of a finite poset / Alg. Universalis 17:73~86
R. Dilworth / 1950 / A decomposition theorem for partially ordered sets / Ann. of Math 51:161~166
R. Dilworth / 1960 / Some combinatorial problems on partially ordered sets / Proc. Sympos. Appl. Math 10:85~90
G. Gratzer / 2011 / Lattice theory: foundation / Birkhauser/Springer Basel
K. M. Koh / 1983 / On the lattice of maximum-sized antichains of a finite poset / Alg. Universalis 17:73~86
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