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So in regular algebra we have some basic operations defined such as multiplication, addition, subtraction and division.

For these operations/operators, we have some properties like commutativity, associativity and distributivity.

Is there an analogue set of properties for the operations/operators on that we do with Regular Expressions?

I also believe that the set of all regular expressions together with its operations that are defined on the members of regular expression set form an algebraic structure similar to the integers and common operators that we learn about at arithmetics.

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  • $\begingroup$ @D.W. OK I will edit the question to match the one in the title and then I will ask another which would be more focused on the existence of various mechanical ways of "minimizing" a regular expression. $\endgroup$
    – yoyo_fun
    Nov 26, 2016 at 23:37
  • $\begingroup$ The answer is yes, but a compendium of all such properties seems to me like it would be too broad. What do others think? $\endgroup$
    – D.W. ♦
    Nov 27, 2016 at 5:06

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There are several known axiomatizations of the equational theory of Kleene algebra, see lecture notes from a course by Dexter Kozen. These axiomatizations give you a complete set of rules that can be used to deduce any identity involving regular expressions. The first ones were given in a classic paper of Salomaa, and others are also known.

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