Pseudocomplement

<h2 id="properties">Properties</h2>
In a p-algebra L, for all 
 
 
 
 x
 ,
 y
 ∈
 L
 :
 
 
 {\displaystyle x,y\in L:}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<ul><li>The map x ↦ x* is <a href="/facts/Antitone/MjQK0YL2">antitone</a>. In particular, 0* = 1 and 1* = 0.</li>
<li>The map x ↦ x** is a <a href="/facts/Closure_operator/Ysc3N0co">closure</a>.</li>
<li>x* = x***.</li>
<li>(x∨y)* = x* ∧ y*.</li>
<li>(x∧y)** = x** ∧ y**.</li></ul>
The set S(L) ≝ { x** | x ∈ L } is called the skeleton of L. S(L) is a ∧-<a href="/facts/Subsemilattice/kW5po0rp">subsemilattice</a> of L and together with x ∪ y = (x∨y)** = (x* ∧ y*)* forms a <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebra</a> (the complement in this algebra is *).<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> In general, S(L) is not a <a href="/facts/Sublattice/5ak6ufky">sublattice</a> of L.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In a <a href="/facts/Distributive_lattice/97DFz1cu">distributive</a> p-algebra, S(L) is the set of <a href="/facts/Complemented_lattice/KpwmsACp">complemented</a> elements of L.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Every element x with the property x* = 0 (or equivalently, x** = 1) is called dense. Every element of the form x ∨ x* is dense. D(L), the set of all the dense elements in L is a <a href="/facts/Filter_(mathematics)/7f3zk55u">filter</a> of L.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> A distributive p-algebra is Boolean if and only if D(L) = {1}.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Pseudocomplemented lattices form a <a href="/facts/Variety_(universal_algebra)/vXWYFyDI">variety</a>; indeed, so do pseudocomplemented semilattices.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="examples">Examples</h2>
<ul><li>Every finite <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a> is pseudocomplemented.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>Every <a href="/facts/Stone_algebra/6tm5YvfE">Stone algebra</a> is pseudocomplemented. In fact, a Stone algebra can be defined as a pseudocomplemented distributive lattice L in which any of the following equivalent statements hold for all 
 
 
 
 x
 ,
 y
 ∈
 L
 :
 
 
 {\displaystyle x,y\in L:}
 
<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
<ul><li>S(L) is a sublattice of L;</li>
<li>(x∧y)* = x* ∨ y*;</li>
<li>(x∨y)** = x** ∨ y**;</li>
<li>x* ∨ x** = 1.</li></ul></li>
<li>Every <a href="/facts/Heyting_algebra/Rrny2LRq">Heyting algebra</a> is pseudocomplemented.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>If X is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, the (open set) <a href="/facts/Topology/2EqW47cU">topology</a> on X is a pseudocomplemented (and distributive) lattice with the meet and join being the usual union and intersection of open sets. The pseudocomplement of an open set A is the <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> of the <a href="/facts/Set_complement/yWUePIYY">set complement</a> of A. Furthermore, the dense elements of this lattice are exactly the <a href="/facts/Dense_set/nPT9Yxao">dense open subsets</a> in the topological sense.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li></ul>
<h2 id="relative-pseudocomplement">Relative pseudocomplement</h2>
A relative pseudocomplement of a with respect to b is a maximal element c such that a∧c≤b. This <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> is denoted a→b. A lattice with the pseudocomplement for each two elements is called implicative lattice, or Brouwerian lattice. In general, an implicative lattice may not have a minimal element. If such a minimal element exists, then each pseudocomplement a* could be defined using relative pseudocomplement as a → 0.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Topological_vector_lattice/q5BLo3cr">Topological vector lattice</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Balbes, Raymond; Horn, Alfred (September 1970). "Stone Lattices". Duke Math. J. 37 (3): 537–545. doi:10.1215/S0012-7094-70-03768-3. <a href="/wiki/Alfred_Horn" target="_blank">/wiki/Alfred_Horn</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">T.S. Blyth (2006). Lattices and Ordered Algebraic Structures. Springer Science & Business Media. Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119. ISBN 978-1-84628-127-3. <a href="978-1-84628-127-3" target="_blank">978-1-84628-127-3</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 63–70. ISBN 978-1-4398-5129-6. <a href="978-1-4398-5129-6" target="_blank">978-1-4398-5129-6</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Birkhoff, Garrett (1973). Lattice Theory (3rd ed.). AMS. p. 44. <a href="/wiki/Garrett_Birkhoff" target="_blank">/wiki/Garrett_Birkhoff</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Pseudocomplement open-in-new

Pseudocomplement