ONT Lattices, Partial Orders, Pre-Orders
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| Lattice.
|
| When x and y are elements of an ordered set L,
| the supremum and infimum of {x, y}, whenever
| they exist, are called the "join" and "meet"
| of x, y and denoted by x |_| y and x |^| y,
| respectively. L is called a "lattice" (or
| "lattice-ordered set") when every pair of
| its elements has a join and a meet.
|
| The following three laws hold in any lattice L:
|
| 1. x |_| y = y |_| x,
|
| x |^| y = y |^| x,
|
| ("commutative law");
|
| 2. x |_| (y |_| z) = (x |_| y) |_| z,
|
| x |^| (y |^| z) = (x |^| y) |^| z,
|
| ("associative law");
|
| 3. x |_| (y |^| x) = (x |_| y) |^| x = x,
|
| ("absorption law").
|
| EDOM 2, page 894.
|
|'Encyclopedic Dictionary Of Mathematics', 2nd edition,
| edited by the Mathematical Society of Japan,
| MIT Press, Cambridge, MA, 1993.
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