The name written in Greek letters: Αβδoυλλα Αρσλαν Βαταιδες, Καισαρ --> Abdoulla Arslan Bataides, Kaisar
With the importance of Arabic in Islam, it makes me wonder how long the alphabet is going to survive or if an Ajami script is going to win out. Perhaps the Greeks will use both scripts and also maybe even the Hebrew Alphabet assuming the Bataid realm has a sufficient Jewish population. I can see Jews from the HRE and Central Europe migrate to Constantinople. This way you have Greek being written in three different scripts at the same time. Now that would be really interesting.
There is an Ajami script for Greek that's bouncing around parts Anatolian at the moment that's become popular among Anatolian Greek conversos. Much like how Andalusian conversos wrote Romance in the Arabic script, so is the case among Greeks who have converted. Arabic - albeit laced with loanwords from Greek and other regional languages - is the language of state, but individual groups still have their own languages, and the historic prominence and longevity of Greek are ensuring that it's likely to stick around.
Ajami Greek would have some precedent-- Jalaleddin Rumi and his son Sultan Walid both lived in the Sultanate of Rum, and while trying to recruit for their Sufi tariqa they actually penned some verses of Cappadocian Greek in an Arabic script. Sultan Walid wrote a lot more (he was born and raised in Rum), and some of his poems are hilariously apologetic about not being very good at Greek even if he is, at least as far as Anatolian Greek conventions go, not that bad. Though there are some scholars who suggest this humility is manufactured-- "I'm not very good at Greek, so why don't you learn Persian and
then I can tell you
all about Sufism," or something like that.
But while it may end up being the most visible script in Anatolia, I don't think it would "win out" and displace the Greek alphabet entirely.
Every language in an Islamic land ended up adopting Arabic script. From Persian to Fulani to Malay and even Uyghur. Greek shouldn't be any different.
The introduction of Latin scripts in some places was only a result of colonialism and European global hegemony.
Not necessarily. Although this rule is near universal there is one exception in the Indonesian islands. The Javanese, Sundanese, and Makassarese all seem to have kept their pre-Islamic scripts and used them in royal chronicles and other texts well into the early 1800s, while Arabic was restricted to explicitly religious texts. Both ended up displaced by Latin, but neither had entirely displaced the other before then (which probably actually just made it easier for Latin to gain predominance).
My personal guess for why this happened is 1) the Javanese and friends had an existing tradition of writing, and 2) the conversion of Java took place by evangelism and then conquest of indigenous actors by indigenous actors, not conquest from outside. The question of what script becomes a society's main secular script seems to depend heavily (though not overwhelmingly) on the writing commissioned by chanceries and bureaucracies, because whatever script they choose to work in will be the one that educated men seeking a job will have to learn whether they want to or not-- and then popular literature will bear the impact of that. The conquests of India (both waves, the Ghaznavid/Ghorid and Mughal) featured a wholesale replacement of royal courts by a new elite of Afghans/Turks, and the institutions of administration saw a rapid influx of Persians-- the pressure to use an Arabic script is doubly strong. And so, in Punjab even the Sikhs used it in secular/governance functions even after they invented Gurmukhi for the specific purpose of avoiding Persian, and even after they became the local elite themselves.
Gurmukhi didn't break out of being a "religious script" until a generation after independence. By contrast, it seems like the courts before and after conversion in Java were composed of
the same people, if that makes any sense, but with a different faith. So there'd be no en masse shift of state preference to a different script, meaning the only partisans for mass Arabic adoption are religious institutions, at which point the script is likely to simply be perceived as a religious script that is "too good" for everyday purposes. Of course this doesn't explain Malaysia's preference for Jawi, but there it seems like some outlying states actually had a heavy Arab influence (the founder of Sulu was an Arab adventurer, Brunei invited a
sayyid to be its third sultan) so I still think I'm right.
Now, what does this mean for the Greeks? Well, Converso Greek is already confirmed to be in Ajami, and this is a believable development-- but they're a minority as of now. The royal/military elite is mostly non-Greek holdovers from the Mamlakate so there's a point for Ajami Greek. The bureaucracy is probably heavily staffed with Persians who migrated in earlier, so there's another point. But as the flow of Persian bureaucrats is choked off by domestic demand in the Mezinids' new conquests and local Greeks start to be more trusted, the decline of Hellenic use may be stalled somewhat. Meanwhile, Hellenic competency will still be in demand-- much as the migration of Byzantine scholars to the west led to a proliferation of Greek texts and of students wanting to read them, similar pressures would be present in the Bataid realm as scholars seek to understand the texts and teachers that stayed behind (though there may not be many left, with Constantinople being devastated so thoroughly TTL. Maybe they're still around in Thessaloniki?). But then again, this didn't do Sanskrit a whole lot of good-- the Mughals simply translated the texts they wanted to read and now Sanskrit competence isn't needed anymore. But to counter that, I think the Muslims are likely to seek out Hellenic knowledge much more fervently than they did Indian knowledge-- the backbone of Islamic medicine is Greek, Pakistani traditional medicine is literally called Tibb-i Yunani (Ionian Medicine). So while the Mughals had a brief burst of Sanskrit-mania that petered out before it could reach the real hidden gems like the Arthasastra, the Bataids' push for fostering Greek-language scholarship might be more sustained, and you may see the script of Bataid medicine/science being not Ajami, but Hellenic (easier to represent Chinese scientific loanwords in an alphabet than an abjad/abugida, maybe?). And as royals lean more heavily on being successors to the Romans, I can see a tradition of Javanese-style royal chronicles in Hellenic. And while some may find this sacrilegious, the Bataids have a pet Caliph to explain why it isn't.
So on the whole, the Hellenic script is probably going to give up a
lot of ground, especially as the Muslim population grows, and after a certain point it may be the clear junior partner to Ajami (or would that be
Atzamiká? Or Açamiká? Or
Êçêmiká?). But the Christians will probably stick with Hellenic for as long as possible, and even Bataid Muslims have an incentive to learn it and the ability to defend their usage of it. So you'd probably have more script diversity than in Andalus-- instead of having Arab domination and native domination be divided by a domination of Saqaliba who have no connection to pre-Islamic scripts, the Bataids kinda skipped that step.
There may also be regional differences in script proportions-- the Peloponnesus might near-universally use Hellenic, while East Anatolia is the reverse.
If that were to happen, I can imagine an offshoot faith that tries to unite all of the Abrahamic faiths into one religion or call itself a purer form of it, sort of like the Baha'i OTL which uses Greek as a unifying language with writing scripts of the three faiths used.
Eleusinian-Twelver Gang? You know, given the Shi'i undercurrent among the Atropatene Turks (Turkish Alevism, the Qizilbash) that finally culminated in the Safavids I'm sure there's something you could do with Shiism in Greece. And then you'd just need some Greek claiming to be [insert End-Of-Days figure] and it's Baha'i time.