Free Dynamic and Algebraic Semantics

Adamek has recently given general criteria for the construction of a free X-dynamics. We specialize this result to the case in which the free dynamics is over the initial object, noting that the result, μ0:A X → A is an isomorphism. This result not only specializes to the theory of minimal fixed points, but provides a new method of constructing solutions to Scott's domain equations which does not require coincidence of limits and colimits. Finally, we show how free dynamics allow us to construct semantics for programming schemes.