Boolean prime ideal theorem

In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. Among the most popular statements of this form is the Boolean prime ideal theorem, which 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, e.g. rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article currently focuses on prime ideal theorems from order theory.

Related Topics:
Mathematics - Abstract algebra - Ideals - Boolean algebra - Filters - Ultrafilter lemma - Rings - Prime ideal - Distributive lattice - Order theory

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Although the various prime ideal theorems may appear simple and intuitive, they can in general not be derived from the axioms of Zermelo-Fraenkel set theory (ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others, like the Boolean prime ideal theorem, represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF+AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.

Related Topics:
Zermelo-Fraenkel set theory - Axiom of choice

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