Editing Heyting algebra (section)

==Quotients==
Let ''H'' be a Heyting algebra, and let {{nowrap|1=''F'' ⊆ ''H''.}} We call ''F'' a '''filter''' on ''H'' if it satisfies the following properties:
#<math>1 \in F,</math>
#<math> \mbox{If } x,y \in F \mbox{ then } x \land y \in F,</math>
#<math> \mbox{If } x \in F, \ y \in H, \ \mbox{and } x \le y \mbox{ then } y \in F.</math>

The intersection of any set of filters on ''H'' is again a filter. Therefore, given any subset ''S'' of ''H'' there is a smallest filter containing ''S''. We call it the filter '''generated''' by ''S''. If ''S'' is empty, {{nowrap|1=''F'' = {1}.}} Otherwise, ''F'' is equal to the set of ''x'' in ''H'' such that there exist {{nowrap|1=''y''<sub>1</sub>, ''y''<sub>2</sub>, ..., ''y''<sub>''n''</sub> ∈ ''S''}} with {{nowrap|1=''y''<sub>1</sub> ∧ ''y''<sub>2</sub> ∧ ... ∧ ''y''<sub>''n''</sub> ≤ ''x''.}}

If ''H'' is a Heyting algebra and ''F'' is a filter on ''H'', we define a relation ~ on ''H'' as follows: we write {{nowrap|1=''x'' ~ ''y''}} whenever {{nowrap|1=''x'' → ''y''}} and {{nowrap|1=''y'' → ''x''}} both belong to ''F''. Then ~ is an [[equivalence relation]]; we write {{nowrap|1=''H''/''F''}} for the [[quotient set]]. There is a unique Heyting algebra structure on {{nowrap|1=''H''/''F''}} such that the canonical surjection {{nowrap|1=''p''<sub>''F''</sub> : ''H'' → ''H''/''F''}} becomes a Heyting algebra morphism. We call the Heyting algebra {{nowrap|1=''H''/''F''}} the '''quotient''' of ''H'' by ''F''.

Let ''S'' be a subset of a Heyting algebra ''H'' and let ''F'' be the filter generated by ''S''. Then ''H''/''F'' satisfies the following universal property:
:Given any morphism of Heyting algebras {{nowrap|1=''f'' : ''H'' → ''{{prime|H}}''}} satisfying {{nowrap|1=''f''(''y'') = 1}} for every {{nowrap|1=''y'' ∈ ''S'',}} ''f'' factors uniquely through the canonical surjection {{nowrap|1=''p''<sub>''F''</sub> : ''H'' → ''H''/''F''.}} That is, there is a unique morphism {{nowrap|1=''{{prime|f}}'' : ''H''/''F'' → ''{{prime|H}}''}} satisfying {{nowrap|1=''{{prime|f}}p''<sub>''F''</sub> = ''f''.}} The morphism ''{{prime|f}}'' is said to be ''induced'' by ''f''.

Let {{nowrap|1=''f'' : ''H''<sub>1</sub> → ''H''<sub>2</sub>}} be a morphism of Heyting algebras. The '''kernel''' of ''f'', written ker ''f'', is the set {{nowrap|1=''f''<sup>−1</sup>[{1}].}} It is a filter on ''H''<sub>1</sub>. (Care should be taken because this definition, if applied to a morphism of Boolean algebras, is dual to what would be called the kernel of the morphism viewed as a morphism of rings.) By the foregoing, ''f'' induces a morphism {{nowrap|1=''{{prime|f}}'' : ''H''<sub>1</sub>/(ker ''f'') → ''H''<sub>2</sub>.}} It is an isomorphism of {{nowrap|1=''H''<sub>1</sub>/(ker ''f'')}} onto the subalgebra ''f''[''H''<sub>1</sub>] of ''H''<sub>2</sub>.