I am a Software Architect and not very familiarized with standard notation in mathematics. Nonetheless, I would like to write a paper explaining a normalization of a computing model for expert systems. It has a very deep background on geometry, logic and group theory, so I have to define some [new] unusual mathematical objects, and in order to get it accepted by the reader (some of them scientists of different disciplines) I would like to be as clear and correct as possible. How can I improve these definitions? Does anyone or any company offer this "help" as a service?

I have about 20 definitions like the following (only for instance) to be improved.

The $m$-crown of a set $S$, denoted by $S^m$, is the family of sets of every subset of its index set of cardinality $m$ not containing its index, such that

$$\forall(X_i \in S^m : i \in S) \rightarrow X_i = \{ x \subset S :(i\notin x \land|x|=m) \}$$

(Note: I am sure that it is a correct definition but may be not easy to understand with a not very standarized notation)