Abstract
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.
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The author acknowledges funding from the National Research Foundation of South Africa through a research grant with grant number 113829.
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Dube, T. Amenable and Locally Amenable Algebraic Frames. Order 37, 509–528 (2020). https://doi.org/10.1007/s11083-019-09517-z
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DOI: https://doi.org/10.1007/s11083-019-09517-z