The principle of arvus means that person a who has already fulfilled his/her (assuming arvus applies to women, see previous posts) mitzvah can still be motzi someone else, e.g. someone who has already said kiddush may recite kiddush for someone who has not. The gemara (Brachos 48) qualifies this with respect to birchas hamazon – the motzi has to have eaten a k’zayis of food. Rashi and Tosfos both question how this din works. The gemara elsewhere (Brachos 20) tells us that someone who is not a bar chiyuva d’oraysa, e.g. a katan, cannot be motzi a man who is mechuyav m’doraysa in bentching. Arvus seems to require parallelism between the chiyuv of the person being yotzei and the person being motzi him/her. How then can someone who ate only a k’zayis, who is only mechuyav mderabbanan in bentching, be motzi someone who ate to the point of satiation, k’dei seviya, who is chayav min hatorah - why is this case different than the case of a katan who cannot be motzi a gadol who ate k'dei seviya?
The simplest answer to this question is that of the BH”G, who denies the whole premis. Perhaps one who ate a k’zayis cannot be motzi someone who ate k'dei seviya - perhaps he can be motzi only someone else who ate a k’zayis! Rashi and Tosfos both reject this chiddush and offer other answers. Rashi explains that a katan is not a bar chiyuva at all – the obligation of chinuch rests entirely on his father (see previous discussion), but someone who ate a k’zayis becomes at least a bar chiyuva on a derabbanan level. Tosfos points out that this begs the question of how a katan can ever be motzi someone considering that he is never a bar chiyuva. Tosfos argues that chinuch does create a chiuv on the child m’derabbanan to do mitzvos, but a katan is still categorically different than a gadol who ate a k’zayis. A katan can never be rise to the level of bar chiyuva d’oraysa no matter how much he eats; a gadol has the potential to be a bar chiyuva d’oraysa if he eats the necessary shiur.
I am still mulling over in my brain what the hesber of the machlokes BH”G and Rashi and Tosfos is and what it tells us about how arvus works (any suggestions?), but for now I wanted to get back to the issue of arvus for women. R’ Akiva Eiger points out that the gemara (20b) lumps together women with ketanim (assuming their chiyuv in bentching is derabbanan). According to Rashi a katan may not have an independent chiyuv derabbanan to bentch, but a woman certainly does – if so, why should the principle of arvus not allow her to be motzi a gadol?! To play devil's advocate, this is only a problem l'shitaso of R' Akiva Eiger, but the Dagul m'Revava might take it as a proof that he is correct in excluding women completely from the principle of arvus.
(For more on the issue of whether a katan is a bar chiyuva, see Koveitz Shiurim of R’ Elchanan, # 30)