1) Why do people refer to the hebrew word for egg, ביצה, as ביה?
2) When one says ביה isn't he also saying part of G-d's Name?
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The Magain Avraham (O.C. 156) quoting the Yam Shel Shlomo says that it is to not say something disgusting (and it should be said in general, not specifically about the Mesechta name).
The Teferes Yisroel (beginning of the Meschta) doesn't like that explanation and suggest instead it is to avoid confusing it with the word בצע which caused a mistake in Halacha at one point in the Talmud.
The word is ביעא, which is Aramaic for egg, not a reference to Hashem's name.
1) The reason I have heard is that it is a matter of etiquette. בצים is a Hebrew euphemism for testicles, so some people, especially chasidim, are careful to use the Aramaic word for egg instead.
2) No, because the Aramaic word for egg is ביעא with a silent yud.