Gemara

(Beraisa): If a Zar ate a k'Zayis of Terumah [b'Shogeg], he pays Keren v'Chomesh (principal and an added fifth);

Aba Sha'ul says, he pays only if he ate Shavah (the value of a) Perutah.

The first Tana learns from "v'Ish Ki Yochal Kodesh bi'Shgagah." The Shi'ur of Achilah (eating) is a k'Zayis. Aba Sha'ul's learns from "v'Nasan [la'Kohen Es ha'Kodesh]." The Shi'ur of Nesinah (giving) is a Perutah.

Bava Metzi'a 47a (Beraisa): A Kli makes Chalipin, even if it is not Shavah Perutah.

Kerisus 6a (Beraisa - R. Meir): One is liable for putting any amount of Shemen ha'Mishchah on a Zar;

R. Yehudah says, one is liable for a k'Zayis.

(Rav Yosef): R. Meir learns from "Al Basar Adam Lo Yisach" and "va'Asher Yiten Mimenu Al Zar [v'Nichras]";

Just like the Shi'ur for Sichah (anointing) is any amount (we never find a Shi'ur for Sichah), also for Nesinah (which is written regarding Kares);

R. Yehudah learns Nesinah from other Mitzvos of Nesinah. The Shi'ur is a k'Zayis.

Menachos 59b - Version #1 (R. Yitzchak bar Yosef): If any amount of oil was put on a k'Zayis of Minchas Chotei, it is Pasul;

"Lo Yasim" refers to any amount. "Aleha" connotes on a proper Shi'ur;

(R. Yitzchak bar Yosef): If a k'Zayis of Levonah was put on any amount of Minchah, it is Pasul.

"V'Lo Yiten connotes putting a proper Shi'ur. "Aleha" connotes on a proper Shi'ur. This is a second source for this (it says "Aleha" also regarding oil), so it excludes. (One may not put on any amount of a Minchah.)

Version #2 - Question (R. Yitzchak bar Yosef): If any amount of oil was put on a k'Zayis of Minchah, what is the law? Must 'Simah' (of oil) resemble 'Nesinah' [of Levonah, which connotes a k'Zayis]?

This question is not resolved.

Gitin 20a - Suggestion: "He will give" mandates that a Get be worth a Perutah!

Rejection: Perhaps the Torah requires only giving a Get, even if it has no value!

Support (Chachamim of Eretz Yisrael): A Get written on Isurei Hana'ah is valid.

Eruvin 49a (Shmuel): Eruv works through acquisition. (All become partners.)

(Rabah): Eruv works through Dirah. (It is as if all live in one domain.)

Question: What is the difference between these explanations?

Answer: They argue about making an Eruv through a Kli, and less than Shavah Perutah.

Kidushin 3a: Chalipin does not work [to be Mekadesh a woman], because Chalipin works with less than Shavah Perutah. A woman does not Makneh herself (alternatively - is not acquired) for less than Shavah Perutah.

Rishonim

Rif and Rosh (Bava Metzi'a 28a and 4:7): We acquire [through Chalipin] with a Kli, even if it is not Shavah Perutah.

Rambam (Hilchos Mechirah 5:1): All Metaltelim acquire each other. If one traded a cow for a donkey, or wine for oil, even though they are particular about the value, once one did Meshichah or Hagbahah, the other acquired what it was traded for.

Rambam (6): We acquire only with a Kli, even if it is not Shavah Perutah. We do not acquire with Isurei Hana'ah, Peros or coins.

Avi Ezri (Rav E.M. Shach Ztz"l, on Hilchos Ishus 1:2 DH veha'Nir'eh): There are two kinds of Chalipin: Kinyan Sudar (one transfers a Kli for the sake of the Kinyan, and it is returned afterwards), and a swap. Even Rav Nachman, who requires a Kli for Kinyan Sudar, agrees that Peros work for a swap (Tosfos 47a DH Ge'ulah, Rambam 5:1). However, for a swap, it must be Shavah Perutah. The Rambam did not say that Chalipin works with less than Shavah Perutah in the beginning of Perek 5, which discusses a swap, until he discusses Kinyan Sudar. This is because a swap is Kinyan Kesef, but Kinyan Sudar is not.

Ran (Kidushin 1a DH Od): Granted, a woman is not Makneh herself for less than Shavah Perutah. Why is Chalipin Pasul for Kidushin when it is Shavah Perutah? The Kinyan is not disqualified because it works with less than Shavah Perutah, rather, only because women are adamant! The Ramban struggled with this. He holds that the primary text says 'a woman is not acquired for less than Shavah Perutah.' I say that initially the Gemara asked that Chalipin should be included in "Kichah", for it is a part of Kinyan Kesef. Even now that we say that it works with less than Shavah Perutah, perhaps since we require a Kli, and something specific, it is important, even though it is not Shavah Perutah. We find that Modeh b'Miktzas (the oath of partial admission) applies when one claims two Kelim and the Nitva admits to one [even if it is not Shavah Perutah - Shevu'os 40b]. Therefore, we can say that it acquires due to Kinyan Kesef. We rejected this. We know that a woman is not Makneh herself for less than Shavah Perutah, even for a whole Kli. Therefore, the verse does not include Chalipin, for if so it would include even less than Shavah Perutah. Rather, it discusses only money, and not Chalipin.

Tosfos (Bava Metzi'a 47a DH Konim): In Pesachim, we say that Nesinah must be [at least] a Perutah. Also regarding Chalipin, it says Nesinah! Pesachim discusses payment for stolen Hekdesh, therefore, it must be at least a Perutah, like Me'ilah, for it is payment. Giving a shoe [for Chalipin] is not giving money. Likewise, giving a Get is not giving money. It is Kosher even on Isurei Hana'ah. It says Nesinah regarding Terumah, yet one blade of wheat exempts the entire stack! It says Nesinah regarding anointing [with Shemen ha'Mishchah], and one opinion obligates for any amount! Even though it is Me'ilah, this is because it says "Sichah", which connotes any amount. The opinion that obligates for a k'Zayis holds like Chachamim who argue with Aba Sha'ul, who say that Nesinah refers to a k'Zayis, like a Zar who ate Terumah.

Tosfos ha'Rosh (47a DH Konim): If a Zar ate Terumah, he must pay for it. We learn from Me'ilah that the Nesinah must be [at least] a Perutah. Giving Terumah is not payment. We cannot learn this from Me'ilah.

Tosfos (Gitin 20a DH Dilma): "He removed his shoe and gave it to his colleague" [from which we learn Chalipin] need not be with a Perutah, even though it says Nesinah. Similarly regarding Terumah, putting blood of Chatas on the Mizbe'ach, putting [blood of the Asham and oil] on the [ear,] thumb and toe [of a Metzora] has no Shi'ur, even though it says Nesinah. Aba Sha'ul exempts for less than a Shavah Perutah because we learn payment for Terumah from payment for Me'ilah. Both are called payment of Kodesh. [A Gezeirah Shavah] "Chet-Chet" equates them. The one who requires a Perutah for anointing learns from the expression of Nesinah. It is really Simah (putting). The Torah called it Nesinah to uproot the connotation of Simah, which is any amount, and requires an important amount, like "v'Nasan la'Kohen Es ha'Kodesh", which is a k'Zayis according to Rabanan of Aba Sha'ul. We require a k'Zayis of Levonah because the Torah initially said "Simah" [and then changed to Nesinah], or like the Isur of burning Ketores outside [the Mikdash], which is a k'Zayis.

Tosfos (Gitin 39b-40a DH Mishum): We do not say that since a Kli Shavah Perutah acquires, all the more so Peros worth a fortune acquire.

Rashash (b'Emtza Devarav d'Ho'il): In Bava Metzi'a and Kidushin, we say that a Kli even less than Shavah Perutah acquires!

Me'iri (49a DH Nechleku): Shmuel says that Eruv acquires for everyone a share of the Chatzer and houses. Rashi's text says that they argue about a Kli, and something that is not Shavah Perutah, i.e. bread. Less than Shavah Perutah acquires only if it is a Kli. Some texts say that they argue about a Kli less than Shavah Perutah. I disagree. It does not matter whether or not the Kli is Shavah Perutah. We say so regarding Modeh b'Miktzas. The Tosefta says that we acquire with a Kli even if it is not Shavah Perutah. The Ra'avad says that other Metaltelim, e.g. garments and similar things, must be Shavah Perutah. We rule that Eruv is due to Kinyan, like it says (62a) that rental [of a Nochri's property] for less than a Perutah suffices.

Note: I do not understand the Ra'avad. Garments are Kelim. Standard Chalipin is with a Sudar (scarf)!

Poskim

Shulchan Aruch (CM 195:2): We make Kinyan Chalipin only with a Kli, even if it is not Shavah Perutah.