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 (19281974)
“Ye say they all have passed away,
That noble race and brave;
That their light canoes have vanished
From off the crested wave;
That, mid the forests where they roamed,
There rings no hunters shout;
But their name is on your waters,
Ye may not wash it out.”
—Lydia Huntley Sigourney (17911865)
“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)