Gemara

(Mishnah): If two partners divided, and Reuven took 10 sheep and Shimon took nine sheep and a dog, all Reuven's sheep are Pesulim for Korbanos. (Each is a Safek Mechir Kelev. Perhaps it was in exchange for the dog.)

Bechoros 57a - Suggestion: If Yesh Breirah, we should be able to pick one of Reuven's sheep to be in exchange for the dog, and the rest will be Kesherim!

Rejection (Rav Ashi): Indeed, if all the animals are worth the same, we may do so;

The case is, the dog is worth more than one of Reuven's sheep. Each of Reuven's sheep is worth slightly more than each of Shimon's. In all, they compensate for the extra value of the dog. Each is partially in exchange for the dog.

Nedarim 45b (Mishnah): If each of two partners (in a courtyard) vowed not to benefit from the other, neither may enter the courtyard;

R. Eliezer ben Yakov permits them, since each enters his own portion.

46b (Rav Huna and R. Elazar): The Halachah follows R. Eliezer ben Yakov.

Bava Kama 51b (Ravina): R. Eliezer ben Yakov holds that Yesh Breirah. When Shimon enters, he enters his half;

Chachamim hold that Ein Breirah. When Shimon enters, he benefits from Reuven's half.

Yoma 55b (Mishnah - R. Meir): One who buys wine from Kusim (and cannot tithe it before drinking) says 'the two Lugim that I will separate later should be Terumah... and may drink immediately;

R. Yehudah, R. Yosi and R. Shimon say, he must separate the tithes before drinking.

Beitzah 38a: R. Oshaya holds Ein Breirah regarding mid'Oraisa laws. Regarding mid'Rabanan laws he holds that Yesh Breirah.

The Halachah follows R. Oshaya.

Bava Basra 107a (Rav): If brothers divided, and a creditor (of their father) took the portion of Ploni (a brother), the division is void.

Rav considers it as if each brother inherited his proper share. All must pay the debt;

(Shmuel): Ploni lost. He is not compensated.

Shmuel considers the brothers to be like buyers (they allow each other to take a share in place of his proper share) without Acharayus (liability to compensate one whose land is legally taken from him).

The Halachah follows Ameimar, who rules like Rav.

Rishonim

Rif and Rosh (Beitzah 21a and 5:7): The Halachah follows R. Oshaya, who holds that Ein Breirah regarding mid'Oraisa laws, but Yesh Breirah for mid'Rabanan laws.

Rambam (Hilchos Ma'aser 7:1): If one has Tevel mid'Oraisa, he must separate the tithes before drinking.

Ri Korkus: Some say that the Gemara rules like R. Oshaya for mid'Rabanan laws, but the Halachah is Yesh Breirah even for mid'Oraisa laws. We learn from Nedarim (45b) and the Mishnah of wine (the stringent opinion forbids due to concern lest the barrel break). The Rambam disagrees.

Rosh (Nedarim 5:1): We hold that (regarding a Chatzer) the Tana'im argue about Breirah.

Question (Rosh ibid.): Rav Huna rules like R. Eliezer ben Yakov, who permits when there is no Din Chalukah (it is a small Chatzer, so one cannot force his partner to divide), due to Breirah. Nevertheless, when there is Din Chalukah, we do not rely on Breirah, for there is a solution through division. Beitzah 38a rules like R. Oshaya, who says that Yesh Breirah for mid'Rabanan laws, but Ein Breirah for mid'Oraisa laws! We follow Rav Nachman in monetary laws, and he says (Kidushin 42b) that brothers who divide an inheritance are like buyers, because Ein Breirah!

Answer #1 (R. Tam, cited in Rosh): We rule like R. Eliezer ben Yakov, but not for his reason (Yesh Breirah). Rather, we permit Vitur (small benefits that people freely give). The Gemara suggested that we should be able to pick one lamb and say that only it was in exchange for the dog. The question was according to the opinion that Yesh Breirah, even though the Halachah does not follow this opinion. Also elsewhere the Gemara asks according to an opinion that is unlike the Halachah.

Rosh: According to this, if they explicitly forbade each other Drisas ha'Regel, they are forbidden, for they show that they are adamant about it. R. Shimshon cites the Ri, who learns from our Gemara that Yesh Breirah for Torah laws. Also in Bava Basra we rule like Rav, that the division is Batel (because Yesh Breirah). The Gemara rules like R. Yoshiyah, i.e. we hold that Yesh Breirah for mid'Rabanan laws. Really, we hold that Yesh Breirah even for Torah laws, unlike R. Yoshiyah. Rav Nachman (Kidushin 42b) does not mean that they are totally like buyers. Rather, they are like buyers, i.e. they divided with intent to be like buyers regarding pardon (if some shares are worth more than others), Ona'ah and Bitul Mekach.

Rosh (ibid.): If the Chatzer has Din Chalukah, even R. Eliezer ben Yakov forbids. We do not rely on Breirah because there is a solution through division.

Answer #2 (Rashba, cited in Rosh): In normal Breirah, we say that after they divide, it is revealed that each one's share was destined for him from the beginning. Here is different. R. Eliezer permits because each partner acquired the right to walk in all of it. Therefore, his partner cannot forbid this to him, just like one cannot forbid David's property to David. Yesh Breirah here means that whenever one walks in the Chatzer, it is as if he alone acquired the place he steps at that time. This is like a pit of partners, which belongs totally to the one who uses it at the time he uses it.

Ran (45b DH v'Ika): Normally, we hold that Ein Breirah. Here is different, for from the beginning we know that each will acquire it at the time he uses it. Only a minority must be decided later, i.e. at what times each acquires it. In such a case, we hold that Yesh Breirah.

Ran (46 DH Aval Yesh): When there is Din Chalukah, since it is destined to be divided, we cannot say that they bought it on condition that each will acquire it at the time he uses it, for perhaps one will force the other to divide! The Rashba forbids even after division, for we cannot say that each received the portion destined for him. This is like normal Breirah, and we hold that Ein Breirah for mid'Oraisa laws. Even though they are like buyers when they divide, a Mudar may not benefit if he bought from the one forbidden to him, unless someone else owned it in between. However, the Rambam permits after division. It seems that even though Ein Breirah for mid'Oraisa laws, we know that they bought the Chatzer with intent that one cannot forbid the other even after division.

Poskim

Shulchan Aruch (YD 226:1): If two partners in a Chatzer vowed not to benefit from each other, if the Chatzer has Din Chalukah, the vow takes effect. Neither may enter until they divide it.

Shach (2): We do not rely on Breirah to permit, because there is a solution through division.

Gra (OC 413:1): R. Yochanan holds that a Get that depends on Breirah does not disqualify to Kehunah (Gitin 25a). We hold (EH 131:4) that it disqualifies. We must say like Rashi, that we are stringent to say Yesh Breirah. However, in many places we say Ein Breirah to be lenient, e.g. R. Yochanan exempts from Ma'aser Behemah, and the Rambam rules like him! In Bava Basra, we hold that the division is Batel. R. Tam says that this is because Yesh Breirah. We hold like Tosfos (Gitin 26b DH Divrei and elsewhere), that when one explicitly stipulated, we are stringent to say Yesh Breirah even for Torah laws. The Rashba holds that Breirah of a Chatzer is unlike regular Breirah. (Rather, from the beginning each partner acquires the part of the Chatzer he will use at the time he uses it.) In Beitzah 39a, Rav Nachman says that Yesh Breirah regarding water for a well (which is like partners in a Chatzer). He was challenged from the exemption of partners from Ma'aser Behemah (regular Breirah)! We can say that only R. Eliezer ben Yakov distinguishes, but Rabanan hold that it is the usual Breirah. The rule is, Yesh Breirah for mid'Rabanan laws and Ein Breirah for Torah laws, whether to be lenient or stringent. If one stipulates explicitly, or it is sure to be clarified, Yesh Breirah even for Torah laws, even to be lenient. The Rambam (regarding a Chatzer) is difficult. Perhaps he holds like the first answer of Tosfos, but this is difficult according to the Rashba. There are difficulties with the Rambam's rulings about Breirah, like the Yam Shel Shlomo said.