Semidistributivity, prime ideals and the subbase lemma
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We generalize the notions of semidistributive elements and of prime ideals from lattices to arbitrary posets. Then we show that the Boolean prime ideal theorem is equivalent to the statement that if a posetP has a join-semidistributive top element then each proper ideal ofP is contained in a prime ideal, while the converse implication holds without any choice principle. Furthermore, the prime ideal theorem is shown to be equivalent to the following order-theoretical generalization of Alexander’s subbase lemma: If the top element of a posetP is join-semidistributive and compact in some subbase ofP then it is compact inP.
KeywordsPrime Ideal Finite Subset Principal Ideal Great Element Prime Element
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