Editing Heyting algebra (section)

== Examples ==
[[File:Rieger-Nishimura.svg|thumb|right|280px|The [[free object|free]] Heyting algebra over one generator (aka Rieger–Nishimura lattice)]]

<ul>
<li> Every [[Boolean algebra (structure)|Boolean algebra]] is a Heyting algebra, with ''p''→''q'' given by ¬''p''∨''q''.</li>
<li> Every [[total order|totally ordered set]] that has a least element 0 and a greatest element 1 is a Heyting algebra (if viewed as a lattice). In this case ''p''→''q'' equals to 1 when ''p≤q'', and ''q'' otherwise.</li>
<li> The simplest Heyting algebra that is not already a Boolean algebra is the totally ordered set {0, {{sfrac|1|2}}, 1} (viewed as a lattice), yielding the operations:
{|
|-style="vertical-align:bottom"
|
<div>
{| class="wikitable"
|+ <math>a \land b</math>
|-
! {{diagonal split header|''a''|''b''}}
| bgcolor="white" width="30" align="center" | 0
| bgcolor="white" width="30" align="center" | {{sfrac|1|2}}
| bgcolor="white" width="30" align="center" | 1
|-
| bgcolor="white" align="center" | 0
| align="center" | 0
| align="center" | 0
| align="center" | 0
|-
| bgcolor="white" align="center" | {{sfrac|1|2}}
| align="center" | 0
| align="center" | {{sfrac|1|2}}
| align="center" | {{sfrac|1|2}}
|-
| bgcolor="white" align="center" | 1
| align="center" | 0
| align="center" | {{sfrac|1|2}}
| align="center" | 1
|}
</div>
| width="30" | 
|
<div>
{| class="wikitable"
|+ <math>a \lor b</math>
|-
! {{diagonal split header|''a''|''b''}}
| bgcolor="white" width="30" align="center" | 0
| bgcolor="white" width="30" align="center" | {{sfrac|1|2}}
| bgcolor="white" width="30" align="center" | 1
|-
| bgcolor="white" align="center" | 0
| align="center" | 0
| align="center" | {{sfrac|1|2}}
| align="center" | 1
|-
| bgcolor="white" align="center" | {{sfrac|1|2}}
| align="center" | {{sfrac|1|2}}
| align="center" | {{sfrac|1|2}}
| align="center" | 1
|-
| bgcolor="white" align="center" | 1
| align="center" | 1
| align="center" | 1
| align="center" | 1
|}
</div>
| width="30" | 
|
<div>
{| class="wikitable"
|+ {{nobold|{{math|''a''→''b''}}}}
|-
! {{diagonal split header|''a''|''b''}}
| bgcolor="white" width="30" align="center" | 0
| bgcolor="white" width="30" align="center" | {{sfrac|1|2}}
| bgcolor="white" width="30" align="center" | 1
|-
| bgcolor="white" align="center" | 0
| align="center" | 1
| align="center" | 1
| align="center" | 1
|-
| bgcolor="white" align="center" | {{sfrac|1|2}}
| align="center" | 0
| align="center" | 1
| align="center" | 1
|-
| bgcolor="white" align="center" | 1
| align="center" | 0
| align="center" | {{sfrac|1|2}}
| align="center" | 1
|}
</div>
| width="30" | 
|
<div>
{| class="wikitable"
| width="50" align="center" | ''a''
| width="50" align="center" | ¬''a''
|-
| bgcolor="white" align="center" | 0
| align="center" | 1
|-
| bgcolor="white" align="center" | {{sfrac|1|2}}
| align="center" | 0
|-
| bgcolor="white" align="center" | 1
| align="center" | 0
|}
</div>
|}
In this example, note that {{math|size=100%|1= {{sfrac|1|2}}&or;¬{{sfrac|1|2}} = {{sfrac|1|2}}&or;({{sfrac|1|2}} → 0) = {{sfrac|1|2}}&or;0 = {{sfrac|1|2}}}} falsifies the law of excluded middle.

<li> Every [[topology]] provides a complete Heyting algebra in the form of its [[open set]] lattice. In this case, the element ''A''→''B'' is the [[interior (topology)|interior]] of the union of ''A<sup>c</sup>'' and ''B'', where ''A<sup>c</sup>'' denotes the [[Complement (set theory)|complement]] of the [[open set]] ''A''. Not all complete Heyting algebras are of this form. These issues are studied in [[pointless topology]], where complete Heyting algebras are also called '''frames''' or '''locales'''.
<li> Every [[interior algebra]] provides a Heyting algebra in the form of its lattice of open elements. Every Heyting algebra is of this form as a Heyting algebra can be completed to a Boolean algebra by taking its free Boolean extension as a bounded distributive lattice and then treating it as a [[generalized topology]] in this Boolean algebra.
<li> The [[Lindenbaum algebra]] of propositional [[intuitionistic logic]] is a Heyting algebra.</li>
<li> The [[global element]]s of the [[subobject classifier]] Ω of an [[elementary topos]] form a Heyting algebra; it is the Heyting algebra of [[truth value]]s of the intuitionistic higher-order logic induced by the topos. More generally, the set of [[Subobject|subobjects]] of any object ''X'' in a topos forms a Heyting algebra.</li>
<li> [[Łukasiewicz–Moisil algebra]]s (LM<sub>''n''</sub>) are also Heyting algebras for any ''n''<ref>{{Cite journal | doi = 10.1007/s10516-005-4145-6| title = N-Valued Logics and Łukasiewicz–Moisil Algebras| journal = Axiomathes| volume = 16| pages = 123–136| year = 2006| last1 = Georgescu | first1 = G. | issue = 1–2| s2cid = 121264473}}, Theorem 3.6</ref> (but they are not [[MV-algebras]] for ''n'' ≥ 5<ref>Iorgulescu, A.: Connections between MV<sub>''n''</sub>-algebras and ''n''-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) {{doi|10.1016/S0012-365X(97)00052-6}}</ref>).
</ul>