is prohibited.
Function ?• maps a concept (X, Y) from the lattice L into a concept of the resulting
lattice L1 by projecting its intent over the attribute set A1: ?• ((X,Y))=((Y??©A1)',
Y??©A1), where the operator ' upon an extent (respectively an intent) returns an intent
(respectively an extent) in L1.
When A1 represents the intent of an existing concept c in L, then L1 is nothing but
the principal filter of c enriched with the partial order.
The Assembly of two lattices L1=B(O, A1, R1) and L2=B(O, A2, R2) according to a same
set O of objects is a substructure of the direct product of L1 and L2 defined by:
??: L1=B(O, A1, R1) ?— L2=B(O, A2, R2) ?†’ L=B(O, A1???A2, R1???R2).
Function ?? maps a pair of concepts from L1 and L2 into a global concept by an intersection
over their respective extents: ?? (??© (X1,Y1),(X2,Y2)???) = (X1??©X2, (X1??©X2)').
The assembly of lattices can be used not only to ???join??? lattices, but also to define
nested line diagrams as described in the Visualization Mechanisms section.
Figure 2 illustrates the projection of lattice L onto the subset abcd and efghi to get
new lattices L1 and L2. The assembly of L1 and L2 produces lattice L.
The selectioN operation, applied to a lattice L=B(O, A, R), produces a new lattice
in which a subset O1 of O is considered. This operation is dual to the projection and
is given by the mapping ??: L?†’L3= B(O1, A, R3).
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