### Applied Algebra

By algebra I don't mean the usual groups, rings, fields, vector spaces etc of a mathematics course. Algebra is much more general, and involves considering the operations on things, and of course equations between these operations.

In applying algebraic ideas to a new field the first problem is to decide what are the basic operations. You don't see this problem in a classical algebra course because the operations of symmetries (groups), quantities (rings, and fields) have long since been clarified. In a new field the choice is very difficult and a bad choice leads to endless complication and confusion.

I have written about this in earlier posts: the parallel operation on processes seems to me to be misguided; the operations on relations in the algebra called "allegories" also seems to me to be misguided. I have suggested alternatives.

Here is another example from the book by Ferenc Gecseg on Products of Automata (Monographs in Theoretical Computer Science, An EATCS Series). A sequential machine has a set of states A, an input alphabet X and an output alphabet Y. It has also a transition function AxX->A and an output function AxX->Y. Now Gecseg defines a "product of sequential machines" as follows:

Click here for the definition.

The definition looks complicated, but even more complicated is the graphical representation:

This graphical representation suggests that there are much simpler operations, with the property that Gecseg's operation is a derived operation. The real entities are the boxes with wires on the boundaries. The real operations are the series and parallel operations, together with certain wire constants. Gecseg's operation is a derived operation in the algebra Span(Graph).

In applying algebraic ideas to a new field the first problem is to decide what are the basic operations. You don't see this problem in a classical algebra course because the operations of symmetries (groups), quantities (rings, and fields) have long since been clarified. In a new field the choice is very difficult and a bad choice leads to endless complication and confusion.

I have written about this in earlier posts: the parallel operation on processes seems to me to be misguided; the operations on relations in the algebra called "allegories" also seems to me to be misguided. I have suggested alternatives.

Here is another example from the book by Ferenc Gecseg on Products of Automata (Monographs in Theoretical Computer Science, An EATCS Series). A sequential machine has a set of states A, an input alphabet X and an output alphabet Y. It has also a transition function AxX->A and an output function AxX->Y. Now Gecseg defines a "product of sequential machines" as follows:

Click here for the definition.

The definition looks complicated, but even more complicated is the graphical representation:

This graphical representation suggests that there are much simpler operations, with the property that Gecseg's operation is a derived operation. The real entities are the boxes with wires on the boundaries. The real operations are the series and parallel operations, together with certain wire constants. Gecseg's operation is a derived operation in the algebra Span(Graph).

Labels: category theory, computing, mathematics

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