The gemara in Brachos has a safeik whether women are chayavos in birchas hamazon min hatorah or not. The Rosh writes that since this issue is unresolved, a woman may not be motzi a man in bentching because her chiyuv may be derabbanan and his chiyuv is d’oraysa. Asks the Rosh, why should this distinction in the levels of chiyuvim matter? – just like a man who ate a k’zayis (chiyuv derabbanan) can be motzi a man who ate k’dei seviya (chiyuv d’oraysa) because of the principle of arvus (i.e. even if I have no chiyuv, I have to help someone else fulfill their chiyuvim), even if a woman’s chiyuv is derabbanan, she could be motzi a man through arvus?! (Arvus would theoretically work even if the one being motzi his/her friend ate nothing, e.g. someone can be motzi a friend in kiddush without drinking the kos, but there is a special din derabbanan by bentching that the motzi has to have eaten). The Rosh answers with a tremendous chiddush that the principle of arvus does not apply to women. The Dagul m’Revava raises a number of questions all based on this yesod of the Rosh, the most well known (O.C. 271) relating to the mitzvah of kiddush. According to the Magen Avraham, a man is yotzei his mitzvah of kiddush m’doraysa by reciting kiddush hayom in ma’ariv of Shabbos. How can a man be motzi his wife, who is chayeves in kiddush min hatorah, when he has already fulfilled his kiddush m’doraysa and the principle of arvus does not apply to women?!
R’ Akiva Eiger completely disagrees with the Dagel m’Revava’s reading of the Rosh. The Rosh never meant that arvus does not extend to women – what the Rosh meant is in a case where the status of chiyuv is a safeik, like the question of a woman’s obligation to bentch, we do not extend the principle of arvus. Perhaps the reason many women who ordinarily do not daven ma’ariv have the minhag of doing so on Shabbos and Y”T night is to avoid this issue raised by the Dagul m’Revava. Depending on how low you define the threshold for mitzvas kiddush, perhaps a woman wishing her husband “Good Shabbos” suffices on a d’orasya level (R’ Akiva Eiger). Furthermore, the Minchas Chinuch (again, thank you Bill Selliger) challenges the whole premis of the MG”A based on the gemara in Pesachim which says that zechiras yetziyas mitzrayim is an integral part of kiddush, which is lacking in ma’ariv. But unless I am missing something, there is another answer available based on yesterday’s discussion. Even though m’doraysa kiddush can be accomplished with a minimal statement mentioning kedushas shabbos, once Chazal added the obligations of kiddush being said al hakos b’makom seudah, then a failure to meet the derabbanan criteria should also mean the d’oraysa kiyum is not fulfilled either.