Rings of Sets and Preorders
Birkhoff (1937) defined a ring of sets to be a family of sets that is closed under the operations of set unions and set intersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called the same structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form a distributive lattice. The elements of the sets may be given a preorder in which x ≤ y whenever some set in the ring contains x but not y. The ring of sets itself is then the family of lower sets of this preorder, and any preorder gives rise to a ring of sets in this way.
Read more about this topic: Birkhoff's Representation Theorem
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and were instant to explain
the three rings of danger.”
—Anne Sexton (19281974)
“She has got rings on every finger,
Round one of them she have got three.
She have gold enough around her middle
To buy Northumberland that belongs to thee.”
—Unknown. Young Beichan (l. 6164)
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That, once grief sets to growing, grief may rest:
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A prudent grief will not despise such aids.”
—Robert Frost (18741963)