Learn Italian: Si sono lasciati senza aver litigato. - They broke up without having argued.

The past infinitive (aver + past participle) shows that the arguing would have happened before they broke up. Senza aver litigato translates to "without having argued." If you used the present infinitive (senza litigare), it would mean "without arguing" right at the exact moment of the breakup.

Why is the word aver missing its final -e?

In Italian, it is very common to drop the final -e of an infinitive verb to make the pronunciation flow better. Both senza aver litigato and senza avere litigato are grammatically correct, but the shortened version is much more frequent.

Why does the first part of the sentence use sono and end in -i (Si sono lasciati)?

Lasciarsi is acting as a reciprocal verb here, meaning "to leave each other" or "to break up." In compound tenses like the passato prossimo, all reflexive and reciprocal verbs must use the auxiliary essere. Because it uses essere, the past participle lasciati must agree with the plural subject.

Why does the second verb use the auxiliary avere (aver litigato) when the first uses essere (sono lasciati)?