Turing Machine where branches are resolved via arbitrary operator

Alternating Turing Machines output Boolean values and combine the values returned by branches via the any/all operators. Is there a name or theory behind the class of Turing Machines where there is no restriction to the Boolean space and the any/all operators?

For example, I want a machine where terminal states output real values and non-terminal states use the min operator to combine the outputs of branches.

Additionally, are there subclasses of this class? I imagine operators which have certain properties (associativity, idempotence, and especially properties related to ordering or transitivity) would have interesting guarantees regarding interruptibility in the same way that a machine using only the any operator can terminate as soon as it finds one accepting state.