Eruvin, Chapter 1, Mishnah 2. In the previous mishnah we discussed the need to make a mavoi (alleyway) fit for use - to render it permissible to carry in it on Shabbos - and we learned that one of the ways to do so is by means of a korah, a beam laid across the top of the entrance to the mavoi. We discussed the required height of the beam as well as the width of the entrance. In this mishnah, the Mishnah explains exactly what may be used to make the mavoi fit and to permit carrying in it.
The dispute between Beis Shammai and Beis Hillel regarding making a mavoi fit:
"Hechsher mavoi" - that which makes a mavoi fit, so that it is permitted to carry in it. "Beis Shammai omrim: lechi v'korah" - both are required together: a lechi, which is an upright post, and also a korah, a beam laid horizontally across the width of the entrance. "U'Beis Hillel omrim: o lechi o korah" - one of them is enough, and there is no need for both.
At the root of the matter there is a fundamental distinction between these two methods:
Lechi - because of a partition: The lechi is considered the beginning of a partition, or the form of a partition, and therefore the mavoi does not have merely three full walls; rather, the lechi joins them as a fourth wall and turns the mavoi into a genuine reshus hayachid (private domain). To such an extent, that one who throws from a public domain into the mavoi is actually liable on Shabbos.
Korah - because of a distinguishing marker: The korah is nothing more than a distinguishing sign, so that a person will not come to confuse the mavoi with a public domain. Without it one may not carry, but once it has been placed there, it does not turn the mavoi into a genuine reshus hayachid, only into a karmelis in which carrying is permitted. Therefore, one who throws a ball from a public domain into the mavoi is not liable, since by Torah law the mavoi is not considered enclosed by four partitions.
A third opinion, and the words of the student in the name of Rabbi Yishmael:
"Rabbi Eliezer omer: lechayayim" - in his view two posts are required.
And furthermore: "Mishum Rabbi Yishmael amar talmid echad lifnei Rabbi Akiva" - one student said before Rabbi Akiva, in the name of Rabbi Yishmael, "shelo nechleku Beis Shammai u'Beis Hillel al mavoi shehu pachos me'arba amos, shehu o b'lechi o b'korah". That is, regarding a mavoi whose entrance is less than four amos wide, all agree with the position of Beis Hillel, that either a lechi or a korah suffices, and even Beis Shammai concede to this.
"Al mah nechleku? Al rachav me'arba amos v'ad eser" - there they disagreed: Beis Shammai said a lechi and a korah, that both are needed, and Beis Hillel said either a lechi or a korah, that one or the other is enough. Rabbi Akiva, however, disagreed with this understanding of the dispute between Beis Shammai and Beis Hillel, and in his view the two disagreed even when the entrance to the mavoi is less than four amos.
The Gemara's question:
On the face of it, Rabbi Akiva's position is identical to that of the first tanna. For the Mishnah cited in the name of Rabbi Yishmael that Beis Shammai and Beis Hillel do not disagree in a case of less than four amos, yet the first tanna and Rabbi Akiva both appear to be saying that they do disagree even in less than four amos. If so, what is the difference between Rabbi Akiva's position and that of the first tanna?
The Gemara explains that the dispute between them concerns whether any measure at all is required when the entrance to the mavoi is smaller than four tefachim: one holds that nothing is needed, and the other holds that at the very least a lechi or a korah is required. The matter is left unresolved, and the Gemara does not determine which of the two opinions holds which position - whether the first tanna holds that no lechi or korah is needed and Rabbi Akiva is the one who holds that they are needed, or precisely the reverse.
In summary: In this mishnah we learned the dispute between Beis Shammai and Beis Hillel regarding making a mavoi fit - a lechi and a korah, or either a lechi or a korah - and the fundamental distinction between the two: the lechi functions as a partition, which turns the mavoi into a private domain, while the korah serves only as a distinguishing marker, permitting carrying but not making the mavoi into a private domain. We also learned the opinion of Rabbi Eliezer, who requires two posts, the words of the student in the name of Rabbi Yishmael that the dispute applies only to a mavoi wider than four amos and up to ten, Rabbi Akiva's disagreement with this, and the Gemara's explanation of the difference between Rabbi Akiva and the first tanna in the case of an entrance smaller than four tefachim.