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
Famous quotes containing the words rings of, rings and/or sets:
“You held my hand
and were instant to explain
the three rings of danger.”
—Anne Sexton (1928–1974)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)
“A continual feast of commendation is only to be obtained by merit or by wealth: many are therefore obliged to content themselves with single morsels, and recompense the infrequency of their enjoyment by excess and riot, whenever fortune sets the banquet before them.”
—Samuel Johnson (1709–1784)