It can be thought of as a variant of intuitionistic logic with star and with a noncommutative conjunction whose identity need not be the top element.
Action algebras were introduced by Vaughan Pratt in the European Workshop JELIA'90.
However, action algebras have the advantage that they also have an equivalent axiomatization that is purely equational.
The last axiom can be written equivalently as a•(a → a)* ≤ a, a form which makes its role as induction more apparent.
In that sense the above axioms constitute a finite axiomatization of regular expressions.
Redko showed in 1967 that the regular expression equations had no finite axiomatization, for which John Horton Conway gave a shorter proof in 1971.
Such anomalous behavior is not possible in an action algebra, which forces a* to be the least transitive reflexive element.