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In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals , or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.
Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms of Zermelo–Fraenkel set theory without the axiom of choice. Instead, some of the statements turn out to be equivalent to the axiom of choice , while others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT are sometimes used to refer to this additional axiom.
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