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complemented lattice (Definition)

Let $ L$ be a bounded lattice (with 0 and $ 1$), and $ a\in L$. A complement of $ a$ is an element $ b\in L$ such that

$ a\land b=0$ and $ a\lor b=1$.

Remark. Complements may not exist. If $ L$ is a non-trivial chain, then no element (other than 0 and $ 1$) has a complement. This also shows that if $ a$ is a complement of a non-trivial element $ b$, then $ a$ and $ b$ form an antichain.

An element in a bounded lattice is complemented if it has a complement. A complemented lattice is a bounded lattice in which every element is complemented.

Remarks.

  • In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be related, or perspective if they have a common complement. For example, the following lattice is complemented.
    $\displaystyle \xymatrix{ & 1 \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\ a \ar@{-}[rd] & b \ar@{-}[d] & c \ar@{-}[ld] \\ & 0 & }$    

    Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third.

  • If a complemented lattice $ L$ is a distributive lattice, then $ L$ is uniquely complemented (in fact, a Boolean lattice). For if $ y_1$ and $ y_2$ are two complements of $ x$, then
    $\displaystyle y_2=1\land y_2=(x\lor y_1)\land y_2= (x\land y_2)\lor(y_1\land y_2)=0\lor(y_1\land y_2)=y_1\land y_2.$
    Similarly, $ y_1=y_2\land y_1$. So $ y_2=y_1$.
  • In the category of complemented lattices, a morphism between two objects is a $ \lbrace 0,1\rbrace$-lattice homomorphism; that is, a lattice homomorphism which preserves 0 and $ 1$.



"complemented lattice" is owned by CWoo.
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See Also: perspectivity, orthocomplemented lattice, pseudocomplemented lattice, difference of lattice elements, pseudocomplement

Other names:  perspective elements, complemented
Also defines:  related elements in lattice, complement, complemented element

Attachments:
relative complement (Definition) by CWoo
uniquely complemented lattice (Definition) by CWoo
sectionally complemented lattice (Definition) by CWoo
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Cross-references: preserves, lattice homomorphism, homomorphism, objects, morphism, category, Boolean lattice, uniquely complemented, distributive lattice, lattice, antichain, non-trivial element, chain, bounded lattice
There are 32 references to this entry.

This is version 23 of complemented lattice, born on 2005-02-16, modified 2008-04-08.
Object id is 6754, canonical name is ComplementedLattice.
Accessed 5368 times total.

Classification:
AMS MSC: 06B05 (Order, lattices, ordered algebraic structures :: Lattices :: Structure theory)
 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets)
Pending Errata and Addenda
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Lattices and power sets by Schneemann on 2006-02-18 14:08:10
Hi!
A small question:

Which additional properties are necessary/sufficient to make a lattice $(L, <)$ isomorphic to a power set (\Pot(M), \subsetneq)?

Ok, the following properties are necessary:
- complete (every subset has an infimum and a supremum)
- complemented
- distributive

But what set of properties is sufficient?
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