The accessible part of a relation R on a set A is a certain subset
of A. It consists of firstly of the objects with no immediate
predecessors, i.e. initial objects; next the objects whose immediate
predecessors are all initial; next the objects whose immediate
predecessors are all the preceding kinds, and so on. More precisely, it is the
least subset X of A which contains a : A whenever it
includes the set
of all immediate
predecessors of a. The relation is well-founded if the whole set A is
accessible.
When the relation R is given by a triple, the immediate predecessors
of an element a:A are given as an indexed set:
.
One can define a family of sets of well-founded trees
(
for a:A) , the set
representing the proposition that
a is accessible, as the least solution (with respect to pointwise
inclusion) of
A slight generalisation of this construction leads to the notion of
the least accessible closure of a subset of A. First we define a
family of trees which may involve leaves.
Not sure where following concepts belong.
Now the accessible closure of a family of sets
indexed by a:A is
the family:
The following dual notion (I don't know a common name for it) suggests
itself. Again, given a family of sets
indexed by a:A, we
define the family: