Gemara

(Beraisa): A case occurred in which spit from a Tzeduki's mouth landed on the Kohen Gadol's garment. The Kohen Gadol asked his wife, who told him that Tzedukiyos fear Chaverim (and follow their rulings).

Eruvin 69a: David went out (on Shabbos) with a band of fragrances. When he saw R. Yehudah Nesi'ah, he covered it. R. Yehudah Nesi'ah said that David can be Mevatel Reshus (grant rights in his property on Shabbos to all members of the Chatzer, according to R. Yehudah. (He is not like a Nochri.)

69b (Beraisa): If a Yisrael observes Shabbos in the market, he can be Mevatel his Reshus. If he does not observe Shabbos in the market, he cannot be Mevatel his premises, for Chachamim said that a Yisrael can give permission or be Mevatel his premises. Regarding a Nochri, one must rent the premises from him.

Chulin 14a (Mishnah): If one slaughters on Shabbos or Yom Kipur, even though he is Chayav Misah, the Shechitah is Kosher.

Rishonim

Rif (Eruvin 21a): One who is Mechalel Shabbos in the market is like a Nochri, so he cannot be Mevatel Reshus.

Rambam (Hilchos Eruvin 2:6): If a Yisrael is Mechalel Shabbos in public, he is like a Nochri in every way. He cannot be Mevatel his Reshus.

Rosh (Eruvin 6:13): Rabbeinu Meir says that a Tzeduki is like a Nochri, for he transgresses mid'Rabanan laws of Shabbos. This is like the first Tana. Even though R. Meir argues, the Halachah follows R. Yehudah against R. Meir. I disagree, In Nidah (33b) we say that wives of Tzedukim fear the Perushim (Chachamim). Also the Ri says that Tzedukim fear Chachamim and do not transgress in public, so they are not considered Nochrim.

Rosh (14): Even if one is transgresses Shabbos through Isurim mid'Rabanan, he is like a Nochri and he cannot be Mevatel his Reshus. We distinguished between Chilul Shabbos in private and in public, but not between an Isur mid'Oraisa and an Isur mid'Rabanan! Rashi says that he cannot be Mevatel because he transgressed Hotza'ah (transferring from Reshus to Reshus) without an Eruv, i.e. an Isur mid'Rabanan.

Rashi (69a DH b'Chumreta): Chumreta d'Medusha is a ring with a signet of a different material. One may not go out with it on Shabbos. I say that it is a band of fragrances. One who goes out with it on Shabbos is Chayav Chatas, for it is a load. It is not an ornament.

Tosfos (Chulin 14a DH ha'Shochet): How can Shechitah on Shabbos be Kosher if he was Mezid? Shechitah of a Mumar to be Mechalel Shabbos is forbidden! We can say that that is in public, and here we discuss in private. Also, one does not become a Mumar due to one transgression.

Poskim

Shulchan Aruch (OC 385:3): If a Yisrael is a Mumar all to idolatry, or to be Mechalel Shabbos in public, even regarding only an Isur mid'Rabanan, he is like a Nochri. If he is Mechalel only in private, even regarding an Isur mid'Oraisa, he is like a Yisrael, and he can be Mevatel Reshus.

Beis Yosef (DH u'Mah she'Chasav Afilu): Tosfos (69a DH Kan), the Rosh, Mordechai and Rashi hold that one who transgresses mid'Rabanan Isurim of Shabbos cannot be Mevatel Reshus. Tosfos explained that Tzedukim can be Mevatel Reshus because they keep even mid'Rabanan laws of Shabbos, for they fear Chachamim. Nowadays, the Tzedukim transgress mid'Rabanan laws of Shabbos, so they cannot be Mevatel. The Rambam brought the Beraisa verbatim, and did not mention Chilul Shabbos mid'Rabanan. Even if he holds that one who does so is a Meshumad, he would not say so about Tzedukim nowadays, for they are not Mezid. They are like Ohnes; they merely follow their fathers' ways. This is like a baby captured by Nochrim, like he wrote in Perush ha'Mishnayos (Chulin 1:1).

Mishnah Berurah (1): Many Poskim say that even Tzedukim nowadays who are Mechalel Shabbos in public are like Nochrim. Kara'ites nowadays are Mechalel Shabbos in public at least regarding mid'Rabanan laws, so they cannot be Mevatel Reshus.

Beis Yosef (DH u'Mah she'Chasav v'Yisrael): Rashi explains that if he observes Shabbos in the market, even if he is Mechalel Shabbos in private (he can be Mevatel).

Magen Avraham (2): If he was Mechalel Shabbos once, he is not called a Mumar (Hagahos Maimoniyos, Tosfos). See YD 11:2. (It says that if one slaughtered on Shabbos, it is Kosher.)

Pri Megadim (Petichah Koleles 4:1): Some say that a Mechalel Shabbos becomes a Mumar only after three transgressions.

Hagahah (1): It seems that he refers to Tosfos, who says that one does not become a Mumar due to one transgression. Also the Mordechai (Chulin 572) and R. Yerucham (15:5) say so. See Prishah (2:19). However, the Ran (1:2) and R. Yerucham explicitly say that he become a Mumar the second time.

Pri Megadim: It is not clear whether one becomes a Mumar to the entire Torah for Hotza'ah into a Karmelis; see Beis Hillel (on YD 2:5). The Pri Chodosh (YD 2, Kuntres Acharon 2) says so only for mid'Rabanan matters (e.g. to be Mevatel Reshus to allow carrying in a Chatzer), but not for Torah matters (e.g. Shechitah).

Pri Megadim (ibid., 3): It seems that one becomes a Mumar only for Chilul Shabbos, but not for Chilul Yom Kipur (Tosfos Chulin 14a, Beis Yosef YD 11). However, Simlah Chadashah (2:16) infers from EH 123 that Yom Kipur is like Shabbos. Indeed, in EH we say that Chilul of Shabbos or Yom Kipur nullifies a Get mid'Oraisa. See the Chelkas Mechokek and Beis Shmuel (brought below). If one was Mechalel Shabbos through the Lav of Mechamer (making an animal work), it is not clear whether he is a Mumar for everything. Based on Tevu'os Shor (2:26), it would seem not. Why should one who is Mechalel Shabbos be a Mumar to the entire Torah? We say so about a Mechalel Shabbos because he testifies falsely that Hash-m did not rest (after creating the world). This does not apply to Yom Kipur! If we would say that a Mumar is Pasul because he does not write Lishmah, like the Beis Shmuel (123:5) says, and therefore we leave her on her Chazakah and she is still (Vadai) married, the Get is Batel. Perhaps Yom Kipur is severe in the eyes of people, so one who is suspected to transgress something severe is suspected to transgress something lenient, so perhaps he does not write Lishmah. Perhaps he is not a Mumar to the entire Torah. Get Pashut says that he is Vadai a Mumar. This requires investigation.

Kaf ha'Chayim (9): Aruch ha'Shulchan says that one becomes a Mumar also for Chilul Yom Tov.

Mishnah Berurah (4): This is only if he is often Mechalel Shabbos. If he did so only once in public, he is not called a Mumar. Some say that he is a Mumar even for one time. 'In public' means in front of 10 Yisre'elim, or he knows that it became known to them. All this refers to one who sins due to desire (for profit or the result of the Melachah), but if he intends to anger Hash-m, even for other Aveiros, even in private, he is like a Nochri.

Mishnah Berurah (5): He is like a Nochri regarding this. It does not help if he is Mevatel his Reshus. They must rent his premises. Regarding other matters, some Acharonim hold that he is not considered a Mumar due to Chilul Shabbos mid'Rabanan. See Pischei Teshuvah (YD 2:8, brought below), Beis Meir and Gaon Yakov.

Mishnah Berurah (6): If one is ashamed to be Mechalel Shabbos in front of a great person, even if he is Machaneh in front of many people, this is considered in private.

Bi'ur Halachah (DH Oh): The Shulchan Aruch discusses one who transgresses because he casts off the yoke of Shamayim. If he errs and thinks that a matter is permitted, presumably, he is not a Mumar. (This is clear from the Ri in Tosfos DH Hotzi. It seems that even Rashi permits, for if one thinks that it is permitted, he is Shogeg.)

Tif'eres Yisrael (Chulin 1:1): One who was Mechalel Shabbos in public once is like a Mumar, even for Chilul Shabbos mid'Rabanan such as moving Muktzeh or carrying outside the Eruv (Tosfos). Simlah Chadashah is lenient about an Isur mid'Rabanan.

Bi'ur Halachah (DH Afilu): It is clear that Rashi holds that he becomes a Mumar even for an Isur mid'Rabanan. Tosfos, the Rosh, Mordechai, Hagahos Maimoniyos, Ritva and Or Zaru'a agree. Their primary proof is from the case of one who carried from the Chatzer to the Mavuy, which is only an Isur mid'Rabanan. I am unsure whether the same applies to an Isur Muktzeh. Perhaps since it is mere movement, it does not make him a Mumar. The Tif'eres Yisrael is stringent also about Muktzeh. This requires investigation.

Kaf ha'Chayim (8): The Shach rules like those who say that one becomes a Mumar through one Chilul Shabbos. Perhaps regarding Eruv we are lenient like the other Poskim (Machatzis ha'Shekel).

Kaf ha'Chayim (11): The Mumar himself counts among the 10 in front of whom he must be Mechalel.

Kaf ha'Chayim (12): Even though he is like a Nochri, one must rent from him for at least a Perutah, for he is a Yisrael. However, perhaps one may rent from him for less than a Perutah, for Chachamim made him like a Nochri in every respect, since Eruvin is only mid'Rabanan.

Kaf ha'Chayim (13): If the only other resident in a Chatzer is a Mumar, is he totally like a Nochri, and he does not forbid? Or perhaps since (mid'Oraisa) he is still a Yisrael, he forbids. However, either way he can be Mevatel his Reshus.

Shulchan Aruch (YD 2:5): A Mumar to be Mechalel Shabbos in public is like a Nochri. (What he slaughters is a Neveilah.)

Shach (17): Poskim argue about one who served idolatry or was Mechalel Shabbos in public once. The primary opinion holds that he becomes a Mumar. However, we do not forbid that Shechitah itself for idolatry or on Shabbos due to Shechitas Mumar. It seems that Maharshal holds like this.

Pischei Teshuvah (8, citing Mishnas Chachamim): One who transgressed Isurim mid'Rabanan of Shabbos or idolatry is not a Mumar to the entire Torah until he transgressed something with a Chiyuv Misah.

Shulchan Aruch (EH 123:5): If a Get was written or signed b'Mezid on Shabbos or Yom Kipur, or on Yom Tov, she is not divorced, for the witnesses are Pasul mid'Oraisa. If was written on Yom Tov b'Mezid, it is Pasul.

Chelkas Mechokek (11): Yom Tov is unlike Shabbos and Yom Kipur. On Shabbos and Yom Kipur, even if only the writer was Mezid, he is like a Nochri if he wrote in public. On Yom Tov, it is Pasul due to the witnesses. The one who wrote is not like a Nochri. He is merely a Yisrael Rasha. He is not disqualified from writing a Get.

Beis Shmuel (8): The general rule is, on Shabbos and Yom Kipur, if he wrote in public b'Mezid, the scribe becomes a Mumar. Even though we hold that one does not become a Mumar due to one transgression, here every two letters is like another action. If he wrote on Yom Tov, he does not become a Mumar. He becomes a Rasha. This is not a Pesul for writing a Get, but if the witnesses are Resha'im, it is Pasul. In any case the Get is not Batel because it is Kosher through the Edei Mesirah.

Rema: If a Get was written b'Mezid on Yom Kipur or Shabbos, it is Batel mid'Oraisa.