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Primes in Preordered Sets: A Question of Henriksen and Kopperman
Author(s) -
JOHNSON D.,
KIST J.
Publication year - 1993
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1993.tb52521.x
Subject(s) - semigroup , intersection (aeronautics) , mathematics , commutative property , prime (order theory) , ideal (ethics) , combinatorics , discrete mathematics , pure mathematics , philosophy , epistemology , engineering , aerospace engineering
. Let X be a family of nonempty proper subsets of the set R ; for a |Ge R , let X a ={ I |Ge X : a ∉ I ), and denote the collection ( X a : a |Ge R } by X R . Henriksen and Kopperman have shown that if X R is closed under finite intersection, then R can be given the structure of a commutative semigroup so that each element of X becomes a prime semigroup ideal; they then ask if the same conclusion can be drawn if the family X R is merely a base for a topology on X . In this paper, certain notions from the theory of preordered sets are used to produce a large family of examples that respond negatively to the Henriksen‐Kopperman question.