The tail-head factorization

The matrix representation of a diagram can be factored into a product of matrices, generally rectangular, which relate the nodes and links to one another, rather than nodes to nodes or links to links. Using capital letters for links, small letters for nodes, the matrix elements of the factors are generally boolean propositions such as ``node a lies at the tip of link B.'' Then the node linking matrix is the product ,

whilst the link joining matrix is , or

Figure 5 shows a diagram upon which its dual has been superimposed; the factorization for is: 0.4em

Reversing the order of the factors produces : 0.4em

There is a temptation to repeat the process to obtain a series of higher order duals; likewise one wonders about the reverse, whereby a given diagram could be exhibited as the dual of another.

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