Gemara

(Rav Papa): Parutz k'Omed (the area of the gaps equals that of the wall) permits;

(Rav Huna brei d'Rav Yehoshua): It does not permit.

Question (Mishnah): If a caravan camped in a valley and surrounded it with Kelim 10 Tefachim tall, one may carry inside if the width of the gaps does not exceed that of the Kelim;

Inference: If the width of the gaps and Kelim are the same, it is permitted!

This is difficult [for Rav Huna].

16b: The Halachah follows Rav Papa.

Question (Rav Hamnuna): Does Omed Merubah Al ha'Parutz permit Arev (a horizontal Mechitzah)?

Answer (Abaye - Mishnah): The total thickness of the ropes must exceed a Tefach, so the top will be at least 10.

If Omed Merubah [Al ha'Parutz] permits Arev, the thickness could be any amount, the spaces between each rope and the rope [or ground] below it would be [just less than] three, three and four Tefachim!

Objection: The gap of four cannot be from the bottom rope to the ground, for then Gediyim Bok'im Bo [goats penetrate it];

It cannot be between the top two ropes, for then the air above and below the top rope is Mevatel it;

If the middle gap is four, it is Omed Merubah from two sides. Does this permit [unlike we concluded on 10b]?!

Correction: Rather, Rav Hamnuna asks about a mat more than seven Tefachim wide. One carved out three Tefachim and left four [above] and Mashehu [below], and hung it within three of the ground.

(Rav Ashi): He asks about a hanging Mechitzah, like R. Tavla:

i. Question (R. Tavla): Does a hanging Mechitzah permit carrying in a ruined house?

i. Answer (Rav): Chachamim gave this leniency only regarding [a Mechitzah over] water.

Rishonim

Rambam (Hilchos Eruvin 16:17): If all the gaps are less than three Tefachim, it is permitted due to Lavud, even though Parutz Merubah Al ha'Omed.

Rambam (18): Parutz k'Omed is permitted. E.g. if one surrounded with sticks or ropes, and there are less than three Tefachim between them, this is a full Mechitzah, even though it is only Shesi (vertical) or Arev.

Rosh (1:22): Rav Hamnuna's question was not settled. Some say that we are stringent, and Omed Merubah does not permit Arev. The Ra'avad says that since Rav Ashi established Rav Hamnuna to discuss another matter, he holds that clearly, Omed Merubah permits Arev like Shesi, so there was no need to ask about this. This is reasonable.

Ri'az (in Shiltei ha'Giborim 4a-b): We forbid Omed Merubah Al ha'Parutz from two sides, i.e. planks two Amos wide on each side of a gap of three Amos. Rather, the gap is Mevatel the plank.

Hagahos Ashri (1:23, citing Or Zaru'a): The air on both sides is Mevatel a Mechitzah when each air by itself exceeds the Mechitzah. If solid walls on both sides of a gap are together greater than the gap, this is like a wall. We do not say so if the walls on the sides are due to Lavud.

Kaf ha'Chayim (75): It seems that even Or Zaru'a requires only that one plank be solid, but the other plank may rely on Lavud.

Tosfos (16b DH Iy): If the gap of four is in the middle, it is Omed Merubah from two sides. It is not a wall, because the air on both sides is Mevatel it. The Sar mi'Kutzi did not understand why the ropes must be a Tefach thick in all. It suffices to have two ropes of Mashehu (each above a gap just less than three), and the top rope is half a Tefach thick and Mashehu, above a gap of three and a half! Perhaps we forbid lest people come to err.

Question (Magen Avraham 362:16, based on Machatzis ha'Shekel): On 10b, we say that even if we permit Omed Merubah from two sides, the air on both sides can be Mevatel. Why does Tosfos equate these? We must say that he asked why on 16b we say that [surely] Omed Merubah from two sides is not Omed. (Machatzis ha'Shekel - in Tosfos' text, the Gemara asserted this simply.) Above (10b), we said that it is [not simple]! Hagahos Ashri must discuss when there is not more air. This requires investigation [for there is always more air above]! Rather, he holds like Tosfos above (10b DH v'Etzba'ayim), that the air on both sides is Mevatel only when we need to consider the air to be sealed due to Lavud.

Poskim

Shulchan Aruch (OC 362:9): Parutz k'Omed is permitted, whether the wall is Shesi or Arev, as long as there is no gap above 10 Amos.

Beis Yosef (DH Parutz): Parutz k'Omed is permitted. We forbid Parutz Merubah only if the gaps are more than three Tefachim. If each gap is less than three, even though in all they are Merubah Al ha'Omed, each is Lavud. The Gemara, Rambam, Rashi and Hagahos Ashri permit surrounding a caravan with ropes or sticks.

Beis Yosef (DH u'Mah she'Chasav Bein): We cannot resolve Rav Hamnuna's question from a caravan. There, the entire Mechitzah is Omed, due to Lavud. Rav Hamnuna mentioned Omed Merubah because he forbids Parutz k'Omed. We permit Parutz k'Omed, therefore we permit even b'Arev. This is like the Ra'avad, Rosh and Rambam, who rule leniently.

Mishnah Berurah (48): One may carry even opposite the Parutz, for it is considered a full Mechitzah.

Mishnah Berurah (50): No Mechitzah may have a gap of three Tefachim below it. Even a board of seven Tefachim without any gap does not permit, even though Omed Merubah. Even a full wall of 10 Tefachim does not permit if it is three above the ground, for Gediyim Bok'im underneath it. If a wall of six Tefachim is on the ground, and after a gap of three there is another Tefach above, it does not permit. Even though Omed Merubah, the air above and below the Tefach is Mevatel it. Similarly, if there are walls three Tefachim thick, on the ground and seven above the ground, even though Omed Merubah, the gap of four in the middle and the air above is Mevatel the three Tefachim on top. However, if there is a wall three thick on the ground and a wall four thick, six above the ground, the gap of three in the middle does not join with the air above to be Mevatel the four on top, since the four exceed the air below them. The air is Batel to the four! It is as if it is totally sealed. Here, the Omed is Merubah. Parutz k'Omed is never permitted for Arev; see the Bi'ur Halachah (below). The Shulchan Aruch cited the Tur. The Tur did not mention Parutz k'Omed regarding Arev, rather, for Shesi. The Rosh said only that Omed Merubah permits Arev.

Bi'ur Halachah (DH Parutz): Parutz k'Omed cannot permit Arev. If there is a wall three and a half thick at the top (of the 10 Tefachim), and a gap three and a half under it, there must be three Tefachim wall on the ground, for if not, it is not a wall at all.

Note: Even if the three on the ground are not solid, e.g. there is a gap of one, and a plank of two, it is considered solid due to Lavud (Kaf ha'Chayim 73, based on the Bach). If the Bi'ur Halachah discussed when there was a plank of one and a half over a gap of one and a half, there is Parutz k'Omed in the entire 10 Tefachim. However, the Bi'ur Halachah connotes that the bottom three Tefachim are solid. He calls his case Parutz k'Omed because the top plank equals the air below it, even though the majority of the 10 Tefachim are wall.

Bi'ur Halachah: We cannot permit, for the air above up to the sky joins with the air in the middle to be Mevatel the top plank. Only when the plank exceeds the air below, it is not Batel, e.g. the plank is four.

Defense (of Beis Yosef - Chazon Ish Mo'ed 68:17): If there are three Tefachim on the ground, and five Tefachim of air above them, and six Tefachim of wall above them, it is permitted. Even if you will say that we require Omed Merubah within 10, there is Parutz k'Omed, which we permit. What extends above 10 helps to prevent the air on both sides being Mevatel. Even the opinion that requires that the wall from one side exceed the gap, the Bi'ur Halachah's case is permitted if the top plank is four. (It extends above 10 Tefachim, and it exceeds the gap underneath. However, this is not Parutz k'Omed - PF.)

Bi'ur Halachah: We permit Parutz k'Omed even on all four walls. The Mishnah says that the gaps may not exceed what stands, i.e. regarding the entire circumference.