无单位正则环与Banaschewski函数的格配位
Coordinatization of lattices by regular rings without unit and Banaschewski functions
摘要：
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。我们证明了一组结果，包括：（1）每个可补模格都有一个带布尔值域的Banaschewski函数，后者直到同构为止是唯一的。(2)每个von Neumann正则环R（不一定是有单位元的）都有一个从R到R的幂等元的映射e，使得对R中的所有x，y，xR=e（x）R和e（xy）=e（x）e（xy）e（x）. (3)每一个有"Banaschewski迹“（Banaschewski函数概念的弱化）的分段补模格，作为一个中性理想，在同一个拟簇内嵌入到某个补模格中。这特别适用于任何具有可数共尾子集的节补模格。节补模格L是可协调的，如果它同构于von Neumann正则环R的所有主右理想的格L（R）.我们说L有一个大的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. Jónsson在1962年证明了如果L有一个可数共尾序列和一个大的4-框架，那么它是可协调的。证明了具有大4-框架的分段可补模格是可协调的当且仅当它有Banaschewski迹。
英文摘要：
A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: (1) Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. (2) Every (not necessarily unital) von Neumann regular ring R has a map e from R to the idempotents of R such that xR=e(x)R and e(xy)=e(x)e(xy)e(x) for all x,y in R. (3) Every sectionally complemented modular lattice with a ``Banaschewski trace'' (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice. This applies, in particular, to any sectionally complemented modular lattice with a countable cofinal subset. A sectionally complemented modular lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of all principal right ideals of a 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. We prove that a sectionally complemented modular lattice with a large 4-frame is coordinatizable iff it has a Banaschewski trace.
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