All in ooc: (very long post ahead, but I couldn't resist)
I'll have to agree with Faa that the Fyros culture seems to have more of a link with the Greco-Ptolemaic (Ptolemaios means warlike in ancient Greek, go figure
) culture than it would with the Roman. It's emperor's names, the names of it's cities, the public baths, academy and agora, the architecture and city-design (well, mainly Pyr).
The Empire as a political system seems to consist of the same mix: a hereditary rule without divine right (the Emperor instead of a king) with satraps (the governors of each region), a senate elected by the Empire's citizens and an over-expanded bureaucracy.
The difference however lies in the more 'modern democratic model'. All Citizens of or born in the Empire are egligable to vote, whereas in Old Greece's Democracies being able to vote meant you had to be a Citizen. Depending on the city-state that would mean: be male, be of age, had served in the military, no record of crimes, be a free man, had been born in the city-state, ...
Due to their warlike nature it comes as no surprise that the Fyros' culture puts a heavy emphasis on military strength, honour and duty. Add the harsh conditions of the Burning Desert and this attitude seems to breed veterans which I would say, leans heavily towards Spartan influences.
Nonetheless, and perhaps more in line with the Old Greek's interpretation of culture (philosophy, literature, public health) I would say gladiatorial combat (as we know it from the Romans: fights to the death as a form of entertainment) does not seem to fit the Fyros culture. Which does not mean the Fyros can not have sporting matches, but I believe they would be more of a non-lethal form. Honour and duty, duty before honour.
Sidenote: the only reference I have found towards fights to the death concerning Old Greek culture is that of more ritual sacrifices during 'funeral games', where the sacrified were more than likely to have been slaves. It is very likely gladiatorial fights in the Roman Empire were adopted from either the Etruscan and Campanian culture who were supposedly fond of 'bloodshed'.
The Matis culture then seems to be heavily influenced by 14th-16th century Europe; Italy for names and architectural/scientific progress during the renaissance, the Iberian peninsula for it's drive of expansion, conquest and enrichment, France for it's absolutism (including the nobles' intrigues at the court) and divine right.
It is possible to define the whole Matis' culture concept as the following: constant striving to perfection for attaining perfection means salvation; there is nothing as pure and perfect to the 'average' Matis as Jena.
But ofcourse, it's never that simple.
This drive for perfection and the absolutism are the inherent cause of arrogance of the culture, towards other races, to those of lower class, other families, those of different faith ... Which explains the race's conquests, annexing, past slavery, culturally and religously subverting others, but also feuds between families, backstabbing and schemeing on the noble/political level. Those who are strong or in power are graced by Jena, those who are weak, unworthy or careless need to be shown the way, and if needed by force.
Now one may think that this leaves little room for the concept of loyalty, but the truth is that the Matis revere power and competition. Again; those in power are graced by Jena, but Jena helps those who help themselves. Dissobeying those in power is blasphemy, bidding your time to work yourself up is cunning and a step closer to the Matis' ideal. Matis live and fight to win.
With that in mind, the sense of competition and constant internal struggles, I'd say the Matis like fights even more than the Fyros would, it's in their blood. Constantly trying to best eachother, constantly proving their worth and looking for improvement. Wether these fights would be of lethal nature, personally I doubt it, but I will leave it in the middle. Ofcourse 'accidents' can happen ...
So chances are the Arena is to be found there exactly for the Matis' competitive nature.
Perhaps there has been a time the Matis would employ slaves or prisoners of war for gladiatorial fights, but I'm guessing those times are long gone by and the 'average' Matis would consider themselves to be above or 'to civilized' to enjoy those. But that doesn't take away the chance there may be some remnants even these days, carefully hidden from public view. Afterall, we know the Slavers are still employed by some Matis in power or with enough dapper ...