did they fight with each other?
Yes, clashes were common between states and chiefdoms.
Maya civilization
Warfare was prevalent in the Maya world. Military campaigns were launched for a variety of reasons, including the control of trade routes and tribute, raids to take captives, scaling up to the complete destruction of an enemy state.
how far back can we trace the mayan culture As far as circa 2000 BC what period of history is that? The Archaic period where were they located? The civilization encompassed areas of south-eastern Mexico, all of Guatemala and Belize, and the western portions of Honduras and El Salvador. did they practice politics? Yes, there were states and chiefdoms which had fluctuating policies towards one another. did they have societies? The society was hierarchical with a sharp difference between the elite and the commoners.
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Maya civilization [SEP] Warfare
Warfare was prevalent in the Maya world. Military campaigns were launched for a variety of reasons, including the control of trade routes and tribute, raids to take captives, scaling up to the complete destruction of an enemy state. Little is known about Maya military organization, logistics, or training. Warfare is depicted in Maya art from the Classic period, and wars and victories are mentioned in hieroglyphic inscriptions. Unfortunately, the inscriptions do not provide information upon the causes of war, or the form it took. In the 8th–9th centuries, intensive warfare resulted in the collapse of the kingdoms of the Petexbatún region of western Petén.