pseudocomplement

Given an element $a$ in a bounded lattice  $L$, a complement  of $a$ is defined to be an element $b\in L$, if such an element exists, such that

 $a\wedge b=0,\qquad{and}\qquad a\vee b=1.$

If a complement of an element exists, it may not be unique. For example, in the middle row of the following diagram (called the diamond  )

 $\xymatrix{&1\ar@{-}[ld]\ar@{-}[d]\ar@{-}[rd]&\\ a\ar@{-}[rd]&b\ar@{-}[d]&c\ar@{-}[ld]\\ &0&}$

any two of the three elements are complements of the third.

To get around the non-uniqueness issue, an alternative to a complement, called the pseudocomplement of an element, is defined. However, the cost of having the uniqueness is the lost of one of the equations above (in fact, the second one). The weakening of the second equation is not an arbitrary choice, but historical, when propositional logic  was being generalized and the law of the excluded middle was dropped in order to develop non-classical logics.

An element $b$ in a lattice  $L$ with $0$ is a pseudocomplement of $a\in L$ if

1. 1.

$b\wedge a=0$

2. 2.

for any $c$ such that $c\wedge a=0$ then $c\leq b$.

In other words, $b$ is the maximal element  in the set $\{c\in L\mid c\wedge a=0\}$.

It is easy to see that given an element $a\in L$, the pseudocomplement of $a$, if it exists, is unique. If this is the case, then the psedocomplement of $a$ is written as $a^{*}$.

The next natural question to ask is: if $a^{*}$ is the pseudocomplement of $a$, is $a$ the pseudocomplement of $a^{*}$? The answer is no, as the following diagram illustrates (called the benzene)

 $\xymatrix{&1\ar@{-}[rd]\ar@{-}[ld]\\ x\ar@{-}[d]&&y\ar@{-}[d]\\ a\ar@{-}[rd]&&b\ar@{-}[ld]\\ &0&}$

The pseudocomplement of $a$ is $y$, but the pseudocomplement of $y$, however, is $x$. In fact, it is possible that $a^{**}$ may not even exist! A lattice $L$ in which every element has a pseudocomplement is called a pseudocomplemented lattice. Necessarily $L$ must be a bounded lattice.

From the above little discussion, it is not hard to deduce some of the basic properties of pseudocomplementation in a pseudocomplemented lattice:

1. 1.

$1^{*}=0$ and $0^{*}=1$ (if $c\wedge 1=0$, then $c=0$, and the largest $c$ such that $c\wedge 0=0$ is $1$)

2. 2.

$a\leq a^{**}$ (since $a^{*}\wedge a=0$ and $a^{*}\wedge a^{**}=0$, $a\leq a^{**}$)

3. 3.

$a\leq b$, then $b^{*}\leq a^{*}$ (since $a\wedge b^{*}\leq b\wedge b^{*}=0$, and $a\wedge a^{*}=0$, $b^{*}\leq a^{*}$)

4. 4.

$a^{*}=a^{***}$ ($a\leq a^{**}$ by $2$ above, so $a^{***}\leq a^{*}$ by $3$, but $a^{*}\leq a^{***}$ by $2$, so $a^{*}=a^{***}$)

Example. The most common example is the lattice $L(X)$ of open sets in a topological space  $X$. $L(X)$ is usually not complemented, because the set complement of an open set is closed. However, $L(X)$ is pseudocomplemented, and if $U$ is an open set in $X$, then its pseudocomplement is $(U^{c})^{\circ}$, the interior of the complement of $U$.

Remarks.

References

• 1 T.S. Blyth, , Springer, New York (2005).
• 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
• 3 R. Halas, http://www.emis.de/journals/AM/93-34/halas.pshttp://www.emis.de/journals/AM/93-34/halas.ps, Archivum Mathematicum (BRNO) 1993.
• 4 S. Ghilardi, http://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdfhttp://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdf, 2000.
 Title pseudocomplement Canonical name Pseudocomplement Date of creation 2013-03-22 15:47:23 Last modified on 2013-03-22 15:47:23 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 21 Author CWoo (3771) Entry type Definition Classification msc 06D15 Synonym pseudocomplemented algebra Synonym Stone algebra Synonym Stone lattice Related topic BrouwerianLattice Related topic ComplementedLattice Related topic Pseudodifference Defines pseudocomplemented lattice Defines benzene Defines p-algebra Defines pseudocomplemented poset Defines relative pseudocomplement