There are no building prerequisites. The building names next to the cost indicate that you may construct the building for free, without having to fulfill the resource cost. From the rulebook, p. 4:
Free Construction (Chains):
Some structures in Age II and Age III have, to the right of their resource
cost, the name of a structure of a previous age. If the player has built the structure named there during a previous Age, that player can build the structure for free, meaning without having to fulfill the resource cost.
Since Halikarnassos allows you to construct any building in the discard pile for free, anyway, the "chained" card is irrelevant at that point. Nevertheless, you are still forbidden from constructing a duplicate card.