algebraic definition of a lattice
The parent entry (http://planetmath.org/Lattice) defines a lattice as a relational structure (a poset) satisfying the condition that every pair of elements has a supremum and an infimum. Alternatively and equivalently, a lattice can be a defined directly as an algebraic structure with two binary operations called meet and join satisfying the following conditions:
Then is reflexive by the idempotency of . Next, if and , then , so is anti-symmetric. Finally, if and , then , and therefore . So is transitive. This shows that is a partial order on . For any , so that . Similarly, . If and , then . This shows that is the supremum of and . Similarly, is the infimum of and .
Conversely, if is defined as in the parent entry, then by defining
the four conditions above are satisfied. For example, let us show one of the absorption laws: . Let . Then so that , which precisely translates to . The remainder of the proof is left for the reader to try.
|Title||algebraic definition of a lattice|
|Date of creation||2013-03-22 17:39:29|
|Last modified on||2013-03-22 17:39:29|
|Last modified by||CWoo (3771)|