In the first sentence, the adverbial absolute phrase, the ship having arrived, is equivalent to the adverbial prepositional phrase, on the arrival of the ship, and defines the time of the action expressed by the verb embarked.

V. INDIRECT OBJECT

480. The simple predicate may be modified by an indirect object ([§ 105]).

In these sentences, the indirect object me, being equivalent to a prepositional phrase, is an adverbial modifier.

The objective of service ([§ 106]) is also an adverbial modifier.

VI. COGNATE OBJECT

481. The simple predicate may be modified by a cognate object or by a phrase containing such an object ([§ 108]).