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Yoghurt can be made by holding a small quantity of live yoghurt mixed with fresh milk and stirred and kept at a constant temperature (38C or 100F) for up to 12 hours.

Can the mixture be put in an electric machine (already turned on) to keep it at temperature on Shabbos?

It would seem that there is no problem of “cooking (bishul)” because the temperature is below “yad soledes bo”. (Is it the same for “insulating (hatmonoh)”?).

One problem might be the element of “building (boneh)” involved in cheese-making but the yoghurt has very little structure.

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The Ben Ish Chay (year 2, B'shalach 20):

Sometimes people make the aforementioned yogurt before Shabas: they heat milk, put it in a receptacle, and put a little made yogurt (or other binding thing) so this milk will harden and become yogurt. It's done by morning.

But sometimes they see in the morning that it's still milk: it hasn't hardened to become yogurt. So they cover it with cloths and put it in a warm spot to finish it up so it hardens; some also put a piece of bread in it, also to finish it up. It's forbidden to do this on Shabas; rather, it should remain the way it is until it hardens and is done by itself. See Chaye Adam 16:11.

…Someone who adds vinegar or bread (or the like) to milk [on Shabas] to make it into yogurt (لبن in Arabic) — this is forbidden; see Chaye Adam.

Chaye Adam (Shabas) 16:11:
It's forbidden to put vinegar or other things in milk [on Shabas] so it becomes whey, as that is actual choosing.
(As far as I know, milk doesn't become only whey when you add vinegar; I assume the CA means that the milk becomes some product that includes whey. A tip of my hat to Stephen G. Brown for the translation as "whey".)

(Note that Mishna B'rura 319 and Aruch Hashulchan 319 don't seem to discuss yogurt. As always, for practical halacha, consult your own rabbi.)

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