Editing Interior algebra (section)

==== Neighbourhood functions and neighbourhood lattices ====
The topological concept of [[Neighbourhood (mathematics)|neighbourhood]]s can be generalized to interior algebras: An element ''y'' of an interior algebra is said to be a '''neighbourhood''' of an element ''x'' if {{math|1=''x'' ≤ ''y''<sup>I</sup>}}. The set of neighbourhoods of ''x'' is denoted by ''N''(''x'') and forms a [[Filter (mathematics)|filter]]. This leads to another formulation of interior algebras:

A '''neighbourhood function''' on a Boolean algebra is a mapping ''N'' from its underlying set ''B'' to its set of filters, such that:

#For all {{math|1=''x'' ∈ ''B'', max{{mset|''y'' ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} exists
#For all {{math|1=''x'',''y'' ∈ ''B'', ''x'' ∈ ''N''(''y'')}} if and only if there is a {{math|1=''z'' ∈ ''B''}} such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}} and {{math|1=''z'' ∈ ''N''(''z'')}}.

The mapping ''N'' of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function ''N'' on a Boolean algebra with underlying set ''B'', we can define an interior operator by {{math|1=''x''<sup>I</sup> = max{{mset|y ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} thereby obtaining an interior algebra. {{tmath|1=N(x)}} will then be precisely the filter of neighbourhoods of ''x'' in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.

In terms of neighbourhood functions, the open elements are precisely those elements ''x'' such that {{math|1=''x'' ∈ ''N''(''x'')}}. In terms of open elements {{math|1=''x'' ∈ ''N''(''y'')}} if and only if there is an open element ''z'' such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}}.

Neighbourhood functions may be defined more generally on [[semilattice|(meet)-semilattice]]s producing the structures known as [[neighbourhood lattice|neighbourhood (semi)lattice]]s. Interior algebras may thus be viewed as precisely the '''Boolean neighbourhood lattices''' i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.