+0
The Mayans may have transported stones over long distances when they built their temples.
+0
It is part of the verb.
The entire verb is 3 words: may have transported.
The main verb is "transport." The tense, present perfect, uses the helping verb have. "May" is an auxilliary verb which changes the mood from one of certainty to one of possibility.
+0
What function ...?

have has the function of moving the concept may into the past.

The Mayans may transport ... ~ It is possible that the Mayans do transport (now) ...
The Mayans may have transported ... ~ It is possible that the Mayans did transport (then) ...

transported has the function of carrying the basic meaning of the physical relocation of the stones by means of an agent (the Mayans in this case). In other words it's the verb that carries the central meaning of the verb phrase. It has to have the -ed because it occurs after have, the auxiliary verb for perfect tenses.

CJ
Teachers: We supply a list of EFL job vacancies
Comments  

It is a verb

Students: Are you brave enough to let our tutors analyse your pronunciation?
Smoking is forbidden
Can you rebel against the laws of nature?

the function of have transported is MVP main verb phrase

Students: We have free audio pronunciation exercises.

Ugh it's on my kids homework and I don't know either

anonymous

Ugh it's on my kids homework and I don't know either

The Mayans may [have [transported stones over long distances when they built their temples]].

"Have" functions as head of the verb phrase in outer brackets.

"Transported" functions as head of the verb phrase in inner brackets.

Although the outer verb phrase is in the perfect tense, "have" is not a present participle but the plain (infinitive) form. The perfect tense can combine with preterite and present tense but can also occur in clauses without inflectional tense, as it does here.