Birkhoff's Representation Theorem - Rings of Sets and Preorders

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 xy 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)

    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 (1791–1865)

    In the beautiful, man sets himself up as the standard of perfection; in select cases he worships himself in it.... Man believes that the world itself is filled with beauty—he forgets that it is he who has created it. He alone has bestowed beauty upon the world—alas! only a very human, an all too human, beauty.
    Friedrich Nietzsche (1844–1900)