Gemara

37a (Beraisa - R. Meir): If two Kiros are connected, and only one of them is Garuf v'Katum (the coals were swept out, or ashes were put on them), Shehiyah is permitted on it, but not on the other.

Beis Shamai forbid any Shehiyah. Beis Hillel permit only hot water.

R. Yehudah says, Beis Shamai permit Shehiyah of hot water, and Beis Hillel permit even a Tavshil;

38a (R. Chiya bar Aba): If one cooked on Shabbos b'Shogeg, he may eat the food. If he was Mezid, he may not eat it.

If he did Shehiyah, there is no distinction.

(Rabah and Rav Yosef): He permits both cases of Shehiyah;

One who cooked b'Mezid may not eat it, for he did an action [on Shabbos]. One who did Shehiyah b'Mezid may eat it, for he did not cook through an action.

(Rav Nachman bar Yitzchak): He forbids both cases of Shehiyah;

One who cooked b'Shogeg may eat it. We are not concerned lest a Yisrael cook on Shabbos b'Mezid. Shehiyah is forbidden only mid'Rabanan. If Shogeg were permitted [b'Di'eved], people might do so b'Mezid, and claim that they are Shogeg.

Question (Beraisa - R. Meir): If one forgot and left a pot on a Kirah and it cooked on Shabbos:

If he was Shogeg, he may eat it. If he was Mezid, he may not eat it.

This refers to partially cooked food. If it was fully cooked, in either case he may eat it.

R. Yehudah permits water that was fully heated, for it is Mitztamek v'Ra Lo. He forbids a fully cooked Tavshil, for it is Mitztamek v'Yafeh Lo.

Rav Nachman bar Yitzchak can say that the Beraisa discusses before the decree (explained below), and R. Chiya bar Aba's law applies after the decree;

However, according to Rabah and Rav Yosef, if R. Chiya's law is before the decree, he should forbid Mezid. If it is after the decree, he should forbid even Shogeg!

This is left difficult.

(Rav Yehudah bar Shmuel): At first, if one cooked on Shabbos, he could eat it if he was Shogeg, but not he was Mezid. The same [distinction between Shogeg and Mezid] applied to one who forgot [to remove the pot, and was Meshaheh];

Many people did Shehiyah b'Mezid and said that they forgot. Chachamim decreed to forbid even one who forgot.

Contradictions: R. Meir contradicts what he said [in the Beraisa on 37a, that Beis Hillel permit only water], and also R. Yehudah does. (There, he permitted a Tavshil Stam, i.e. even Mitztamek v'Yafeh Lo!)

Answer: Here, R. Meir permits only b'Di'eved. He forbids l'Chatchilah. There, R. Yehudah discusses a Kirah that was Garuf v'Katum. Here he discusses one that was not.

Question: If one transgressed Shehiyah, is the food forbidden?

Answer (Shmuel bar Noson): A case occurred in which R. Yosi saw Shehiyah of water and dried eggs on Kiros. He did not forbid the water, but he forbade the eggs.

Suggestion: He forbade them [b'Di'eved] that very Shabbos!

Rejection: No, he forbade them to do so l'Chatchilah the next Shabbos.

Rishonim

Rif (16b): If one forgot on top of a Kirah that is not Garuf v'Katum, something that is not fully cooked, it is forbidden, and all the more so if he was Mezid. The Gemara asked about one who was Meshaheh a fully cooked food that is Mitztamek v'Yafeh Lo, and never resolved this. This is a Safek mid'Rabanan, so we are lenient.

Rambam (Hilchos Shabbos 3:9): Any Tavshil that one may not be Meshaheh, if one transgressed and was Meshaheh, one may not eat it until Motza'ei Shabbos, after bi'Chdei she'Ya'asu (the time to cook it after Shabbos). If he forgot, if it was not fully cooked, he may not eat it until Motza'ei Shabbos. If it was Mitztamek v'Yafeh Lo, he may eat it immediately on Shabbos.

Rebuttal (Ra'avad): He holds that the Gemara asked about one who forgot [to remove] a Tavshil that is Mitztamek v'Yafeh Lo, and it was not resolved. This is wrong. R. Meir and R. Yehudah argue about this, we hold like R. Yehudah! After the decree, even R. Meir forbids also Shogeg! Rather, the Gemara asked according to R. Yehudah about one who did Shehiyah b'Shogeg (he thought that it is permitted) if it was Mitztamek v'Yafeh Lo. Perhaps he is stringent for one who forgot, but he is lenient about Shogeg if it was Mitztamek v'Yafeh Lo, and he does not fine due to Mezid. This was not resolved, so we are lenient. R. Yehudah found shriveled eggs, and forbade them the next Shabbos. Surely they forgot or were Shogeg. If they forgot, why is the next Shabbos different? Rather, they were Shogeg. He forbade, lest they do so again.

Rosh (3:1): The Gemara asked about one who forgot a pot on top of coals and it cooked on Shabbos. It asks according to Chananyah, and according to Rabanan. It asks according to Chananyah if the food was not cooked k'Ma'achal Ben Drusai before Shabbos, even after the decree. Perhaps the decree was only to the Meshaheh, or to his entire family. We answer that "forbade" connotes to everyone. Alternatively, we asked about a cooked food that is Mitztamek v'Yafeh Lo, but it need not dry out. We learn from the eggs. The Gemara connotes that they were already dry when Shehiyah was done.

Tosfos (38a DH Shachach): Even though we hold like Chananyah, we ask about food that was not k'Ma'achal Ben Drusai before Shabbos. One may not be Meshaheh it. Alternatively, we ask even about a raw Tavshil. Above we permit only in a Tanur, which is very hot. Without stoking it will be ready for the morning. Wind cools down a Kirah, so we are concerned lest he stoke.

Tosfos (DH Avar): The latter question is not about something cooked less than k'Ma'achal Ben Drusai, for we already forbade above! The Ri says that it asks about one who cooked b'Shogeg, according to R. Yehudah. Even though the Halachah does not follow him, we ask whether he forbids Mitztamek v'Yafeh Lo even b'Shogeg. However, "Avar v'Shihah" connotes Mezid. The Rashba says that we ask about before the decree. Afterwards, Shogeg is forbidden like Mezid!

Poskim

Shulchan Aruch (OC 253:1): If one forgot and was Meshaheh, if it was fully cooked, it is permitted even if it is Mitztamek v'Yafeh Lo. If it began cooking but was not fully cooked, it is forbidden until Motza'ei Shabbos. If he transgressed and was Meshaheh, both of these are forbidden.

Beis Yosef (DH Kol): Rashi holds that R. Chiya bar Aba asked about a Tavshil that cooked but did not reach k'Ma'achal Ben Drusai. If it did not cook at all, or it reached k'Ma'achal Ben Drusai, Shehiyah is permitted. According to the Rif and Rambam, he asked also about a food that reached k'Ma'achal Ben Drusai, but it is not fully cooked for most people. If it was fully cooked and Mitztamek v'Yafeh Lo, this is the coming question! The Mordechai (261) and one answer in Tosfos say that we ask about a totally raw food. We permitted it in a Tanur. Perhaps a Kirah is different. We conclude that Chachamim decreed to forbid.

Beis Yosef (DH v'Su): The Rosh says that the latter question was whether the decree was only to the Meshaheh, or to his entire family. We answer that all are forbidden. However, the Mordechai says that only the Meshaheh is forbidden. The Rif's text said that we asked about one transgressed and forgot (in place of transgressed and was Meshaheh. He holds that the question was about food that is fully cooked, and Mitztamek v'Yafeh Lo. Perhaps the decree against one who forgot was only if it was not fully cooked. It was not resolved, so we are lenient. The Rambam agrees. The Tur did not think to explain like this, so he asked about how the Rambam distinguished between transgressing (Shehiyah) and forgetting, for the Gemara equated them! I answer that they are the same for a food that was not fully cooked. The Kol Bo says that the Rif permits Mitztamek v'Yafeh Lo even if he transgressed [b'Mezid] Shehiyah, and all the more so if he forgot. This is astounding. The Rif permits only one who forgot! The Ra'avad's text says that we asked about one who transgressed and did Shehiyah, i.e. b'Shogeg. This is more lenient than one who forgot.

Magen Avraham (11): The Shulchan Aruch wrote Stam forbidden, i.e. to everyone, like the Tur says. Darchei Moshe agrees. The Mordechai permits to others. He holds like R. Meir, but we (318:1) hold like R. Yehudah. Surely it is forbidden to the family of the Meshaheh, since he cooked for their sake.

Gra (32 DH v'Im Shachach): The Mordechai is wrong. We equate Shogeg to Mezid, and Mezid is forbidden to everyone. Rashi (Bava Kama 71a DH Mezid) and Tosfos (Gitin 53b DH ha'Mevashel) say that Mezid of R. Meir is like Shogeg of R. Yehudah.

Mishnah Berurah (29): All the more so we permit if he forgot the Halachah.

Mishnah Berurah (30): We permit Mitztamek v'Yafeh Lo because he did not gain very much, since it was already fully cooked.

Kaf ha'Chayim (21): Several permit Shogeg like one who forgot, but the Aruch ha'Shulchan says that Shogeg is like Mezid.

Kaf ha'Chayim (22): If one cooked b'Mezid, it is forbidden to him forever. Here it is not, for he transgressed only mid'Rabanan.

Rema: They are forbidden until bi'Chdei she'Ya'asu.

Beis Yosef (DH Kol): Hagahos Ashri forbids until bi'Chdei she'Ya'asu on Motza'ei Shabbos. It seems that the Rambam agrees.

Eshel Avraham (11): The Rambam mentions bi'Chdei she'Ya'asu only regarding one who intentionally was Meshaheh, but not regarding one who forgot.

Kaf ha'Chayim (20): The Bach permits even to his family, but the Acharonim forbid even to others bi'Chdei she'Ya'asu.

Shulchan Aruch (ibid.): Some permit Shehiyah of any food that reached k'Ma'achal Ben Drusai, or it was fully cooked and it is Mitztamek v'Yafeh Lo. If one forgot and was Meshaheh, if it began cooking but did not reach k'Ma'achal Ben Drusai, it is forbidden, and we need not teach that it is forbidden if he transgressed and was Meshaheh.

Kaf ha'Chayim (23): The custom is to be lenient like this opinion.

Mishnah Berurah (43): B'Di'eved, a third cooked is considered k'Ma'achal Ben Drusai, for most hold like this.