We say a prime element of an algebraic frame is amenable if it comparable to every compact element. If every prime element of an algebraic frame L is amenable, we say L is an amenable frame. If the localization of L at every prime element is amenable, we say L is locally amenable. These concepts are motivated by notions of divided and locally divided commutative rings. We show that an algebraic frame is (i) amenable precisely when its prime elements form a chain, and (ii) locally amenable precisely when its prime elements form a tree. Given any prime element p of L, we construct a certain pullback in the category of algebraic frames with the finite intersection property on compact elements, and characterize (in terms of the localization Lp and the quotient ↑p of L) when this pullback is amenable and when it is locally amenable.

中文翻译：

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适合的和局部适合的代数框架

如果代数框架的素元与每个紧元可比，我们就说它是适合的。如果代数坐标系 L 的每个素元都是可服从的，我们说 L 是可服从坐标系。如果 L 在每个素数元素上的定位是可服从的，我们说 L 是局部可服从的。这些概念是由划分的和局部划分的交换环的概念驱动的。我们证明了一个代数框架是（i）当它的主要元素形成一个链时精确地服从，并且（ii）当它的主要元素形成一棵树时局部地服从。给定 L 的任何素元素 p，我们在紧元上构造具有有限相交性质的代数框架类别中的某个回拉，并表征（根据 Lp 的局部化 Lp 和商 ↑p）何时该回拉是可接受的并且当它适合当地时。