How can an extended linearizable history G be equivalent to sequential history S?
The idea of linearizability is that the parallel events behave as if each of them happened at a single point in time, during the duration of the actual execution of the event. Given a sequence of events, if the end point of event $A$ precedes the starting point of event $B$ then in any linearization, the time point of $A$ must precede the time point of $B$. These are the constraints $\rightarrow_G$, which we can think of as a partial order. The events are linearizable if the constraints are extendable to a linear order $\rightarrow_S$ such that the semantics match &mdash; the outcome of the events is the same as if they were applied in the order underlying $\rightarrow_S$. Perhaps this is what was meant by equivalence, though as you mention, it is clearly applied to the wrong objects. What is equivalent here is the actual turn of events and the turn of events given the order $\rightarrow_S$.