2 resultados para Ideal lattices
em CaltechTHESIS
Resumo:
Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k) ≥ w(1) for k = 2, 3, ..., n-1. Second, w(k) = w(1) if and only if k = n-1 and L is modular. Several corollaries concerning the "matching" of points and dual points are derived from these theorems.
Both theorems can be regarded as a generalization of a theorem of de Bruijn and Erdös concerning ʎ= 1 designs. The second can also be considered as the converse to a special case of Dilworth's theorem on finite modular lattices.
These results are related to two conjectures due to G. -C. Rota. The "unimodality" conjecture states that the w(k)'s form a unimodal sequence. The "Sperner" conjecture states that a set of non-comparable elements in L has cardinality at most max/k {w(k)}. In this thesis, a counterexample to the Sperner conjecture is exhibited.
Resumo:
A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let M∞n denote the lattice variety generated by all modular lattices of width not exceeding n. M∞1 and M∞2 are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that M∞3 is also finitely based. On the other hand, K. Baker has shown that M∞n is not finitely based for 5 ≤ n ˂ ω. This thesis settles the finite basis problem for M∞4. M∞4 is shown to be finitely based by proving the stronger result that there exist ten varieties which properly contain M∞4 and such that any variety which properly contains M∞4 contains one of these ten varieties.
The methods developed also yield a characterization of sub-directly irreducible width four modular lattices. From this characterization further results are derived. It is shown that the free M∞4 lattice with n generators is finite. A variety with exactly k covers is exhibited for all k ≥ 15. It is further shown that there are 2Ӄo sub- varieties of M∞4.