Algebra of Transformations of Formulas

In the online article Axiomatic Theory of Formulas I described the algebra of formulas, a very generalized abstract way to research any mathematical expressions (including infinite expressions).

Based on that algebra there can be developed a higher-degree algebra, which I call algebra of transformations of formulas. It is probably important for development of post-axiomatic mathematics, because a main object of research of post-axiomatic mathematics is namely transformations of formulas (that is functions on the set of formulas).

A system of formulas, as defined in the above mentioned online article, is essentially the following abstract algebra (with possibly infinite set of operations):

Base set

• any set (called a set of formulas).

Operations

• X (indices) is a set of unary operations;
• Z (null formulas) is a set of nullary operations (most often Z is an one-element set);
• Y (constant symbols) is a set of nullary operations.

Equational laws

• i(0) = 0 for i in X, 0 in Z.

To any system of formulas corresponds the following universal algebra of the operators on the set of formulas:

Base set

• the set of operators, that is functions on the set of formulas (with requirement that these map null formulas to itself).

Operations

• Z* (zeros) is a set of nullary operations (in practice most often Z is a single element set, Z = {0});
• o (composition of operators) is a binary operation;
• [] (so called conditional pass operation) is an unary operation.

Equational laws

• r o (q o p) = (r o q) o p;
• [[p]] = [p];
• [q o p] o [p] = [p] o [q o p] = [q o p];
• p o [p] = p;
• p o 0 = 0 for 0 in Z*;
• [q] o p = p o [q o p].

Additionally we can add the following operations (which are however not related with above introduced operations by any known equational laws):

• X* (the set of indices) is a set of unary operators;
• S* (the set of symbols) is a set of nullary operations.

The issues are: Have I missed any equational laws which could be also added to the system of transformations of formulas? What is exactly the algebra of operators on a set of formulas? How to reflect the set of symbols Y in the higher degree algebra of transformations of formulas?

Please send me any ideas, notes (maybe you have already seen a similar algebra?)

You can also use alt.math.post-axiomatic newsgroup if it is accessible from your news server.

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Author: Victor Porton - www.mathematics21.org