具有大Jónsson四框架的不可协调的分段可补模格
A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame
摘要：
一个分段可补模格L是可协调的，如果它同构于某个von Neumann正则环R的所有主右理想的格L（R）.我们说L有一个大4-框架，如果它有一个齐次序列（a_0，a_1，a_2，a_3），使得a_0生成的中立理想是L。琼森在1962年证明，如果L有一个可数的共尾序列和一个大的4-框架，那么它是可协调的;是否可以免除共尾序列假设是开放的。我们通过寻找一个具有大4-框架的不可协调的分段可补模格L来解决这个问题，它的基数为1。此外，L是具有5-框架的可补模格中的理想。我们的证明使用Banaschewski函数。有界格L上的Banaschewski函数是L的反调自映射，它为L的每个元素选取一个补。在较早的文章中，我们证明了每一个可数可补模格都有一个Banaschewski函数。证明了存在基数为1、指数为3的单位正则环R，使得L（R）不含Banaschewski函数.
英文摘要：
A sectionally complemented modular lattice L is coordinatizable if it is isomorphic to the lattice L(R) of all principal right ideals of some von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. Jónsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented modular lattice L with a large 4-frame; it has cardinality aleph one. Furthermore, L is an ideal in a (necessarily coordinatizable) complemented modular lattice with a spanning 5-frame. Our proof uses Banaschewski functions. A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented modular lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality aleph one and index of nilpotence 3 such that L(R) has no Banaschewski function.
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