An orthocomplemented lattice is a complemented lattice in which every element has a distinguished complement, called an orthocomplement, that behaves like the complementary subspace of a subspace in a vector space.
Formally, let be a complemented lattice and denote the set of complements of elements of . is clearly a subposet of , with inherited from . For each , let be the set of complements of . is said to be orthocomplemented if there is a function , called an orthocomplementation, whose image is written for any , such that
is order-reversing; that is, for any , implies .
The element is called an orthocomplement of (via ).
the one on the right is orthocomplemented, while the one on the left is not. From this one deduces that orthcomplementation is not unique, and that the cardinality of any finite orthocomplemented lattice is even.
From the above conditions, it follows that elements of satisfy the de Morgan’s laws: for , we have
To derive the first equation, first note . Then . Similarly, . So . For the other inequality, we start with . Then . Similarly, . Therefore, , which implies that .
Conversely, any of two equations in the previous remark can replace the third condition in the definition above. For example, suppose we have the second equation . If , then , so , which shows that .
From the example above, one sees that orthocomplementation need not be unique. An orthocomplemented lattice with a unique orthocomplementation is said to be uniquely orthocomplemented. A uniquely complemented lattice that is also orthocomplemented is uniquely orthocomplemented.
Orthocomplementation can be more generally defined over a bounded poset by requiring the orthocomplentation operator to satisfy conditions 2 and 3 above, and a weaker version of condition 1: exists and . Since is an order reversing bijection on , and . From this, one deduces that iff . A bounded poset in which an orthocomplementation is defined is called an orthocomplemented poset.
- 1 G. Birkhoff, Lattice Theory, AMS Colloquium Publications, Vol. XXV, 3rd Ed. (1967).
|Date of creation||2013-03-22 15:50:36|
|Last modified on||2013-03-22 15:50:36|
|Last modified by||CWoo (3771)|
|Defines||uniquely orthocomplemented lattice|