Greatest lower bound, or intersection

Consider the same graphs,

representing a digon, and

representing a triangle. The loops of the first have length 2, those of the second, length 3. It would seem that loops common to the two would have length 6; what is the diagram which would express this fact? The evident candidate is the cartesian product of the two graphs, linkage between its points occurring only if both coordinates were linked, for which we use the symbol .

Effectively, linkage in the cartesian product is defined by

In this example, the intersection has nodes

with links

Graphed by usual conventions, a folded hexagon results, manifesting the required cycle of length 6, as seen in Figure 2.

 
Figure 2: Greatest lower bound of a digon and a triangle.  

The topological matrix of the intersection of two diagrams is the tensor product of the topological matrices of the constituents.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx