Gemara

(Beraisa): Oso v'Es Beno applies to a mother (and her child), but not to a father;

Chananyah says, it applies to mothers and fathers.

He explains that "Oso" (masculine) denotes the father. "Its son" denotes the parent that a child clings to, i.e. the mother. Therefore, both are included.

(Beraisa): Had the Torah said 'Shor v'Seh u'Veno', one might have thought that one transgresses only if he slaughters all three. "Shor Oh Seh Oso v'Es Beno" teaches that this is not so.

Chachamim learn this from "Oso", and "Oh" includes Kil'ayim;

Chananyah uses Oso like above. He holds like R. Yonason, whenever the Torah says "and", it means "or" unless it specifically says "together" (so "Oh" is extra, to include Kil'ayim.)

79a (Shmuel): The Halachah follows Chananyah.

This is consistent with another teaching of Shmuel;

(Mishnah - R. Yehudah): One may mate offspring of a female horse, or make them work together, even if the fathers were donkeys;

A mule born from a (female) horse is forbidden with one born from a (female) donkey.

(Shmuel): This is R. Yehudah's opinion. He is not concerned for the father's seed. Chachamim say that all mules are one species (and are permitted with each other.)

He calls Chananyah's opinion 'Chachamim.' We are concerned for the father's seed. Every mule is a half-donkey, half-horse.

Question: Is R. Yehudah surely not concerned for the father's seed, or is he in doubt?

Answer (Rav Huna brei d'Rav Yehoshua): All forbid a mule with its mother.

This shows that R. Yehudah was in doubt.

(Abaye): If a mule has a thick voice, its mother was a horse. If not, it was born to a donkey.

(Rav Papa): A mule from a (female) donkey has big ears and a small tail. A mule born from a horse is vice-versa.

R. Aba told his servant 'if you lead my wagon with mules, check that they have the same mother.'

Inference: He is not concerned for the father's seed.

79b (Beraisa): Oso v'Es Beno applies to Kil'ayim and to a Koy (a crossbreed of a goat and a deer);

R. Eliezer says, it applies to Kil'ayim from a goat and sheep, but not to a Koy.

Chachamim are concerned for the father's seed. The Torah forbids a Seh and its son, even a partial Seh;

R. Eliezer exempts. We are not concerned for the father's seed, so the Koy is not even a partial Seh.

Question (Beraisa): Matanos (the foreleg, jaw and stomach) of a Koy or (other) hybrids must be given to a Kohen;

R. Eliezer says, Matanos must be given from a crossbreed of a goat and sheep, but not from a Koy.

Granted, Chachamim obligate half Matanos (for it is at least a half-Seh). However, R. Eliezer should obligate full Matanos!

Answer: Really, also R. Eliezer is unsure whether we are concerned for the father's seed. Chachamim hold that a partial Seh is called a Seh, and R. Eliezer disagrees.

If a Koy was born from a male goat and a female deer, all would permit to slaughter it and do Kisuy on Yom Tov. Kisuy applies even to a partial deer (Chayah)!

Kesuvos 111b (Yitzchak, Shimon and Oshaya): The Halachah follows R. Yehudah regarding mules;

(Beraisa - R. Yehudah): We may not mate a mule with a horse or donkey, only with a mule like itself.

Bechoros 45b (Mishnah): Oso v'Es Beno disqualifies animals, but not people.

This means that a Kohen and his children may serve together.

Objection: The corresponding case of animals is permitted!

(Beraisa): Oso v'Es Beno applies to a mother, but not to a father.

Answer (R. Yosi bar Avin): Our Mishnah is like Chananyah.

Rishonim

Rif (Chulin 27a DH Gemara): some say that the Halachah does not follow Chananyah, for R. Aba told his servant 'if you lead my wagon with mules, check that they have the same mother.' He holds that we are not concerned for the father's seed. Yitzchak, Shimon and Oshaya rule like R. Yehudah regarding mules. The Halachah follows R. Yehudah, that we are not concerned for the father's seed. Some say that if we are sure which is the father, we do not slaughter the child and the father on the same day, for Shmuel rules like Chananyah. Presumably, the Halachah does not follow Chananyah.

Rambam (Hilchos Shechitah 12:11): Oso v'Es Beno applies to females, for we know for sure which is its child. If we are sure which is the father, we do not slaughter the child and the father on the same day. If one slaughtered, he is not lashed, for it is a Safek whether it applies to males.

Rosh (5:2): R. Aba is not concerned for the father's seed. He and all the Amora'im who gave Simanim for the two kinds of mules hold like R. Yehudah, who is unsure whether we are concerned for the father's seed. In Kesuvos, Yitzchak, Shimon and Oshaya rule like R. Yehudah regarding mules. The Targum of the verse of Oso v'Es Beno discusses the mother. This implies that Oso v'Es Beno does not apply to males. However, the Stam Mishnah in Bechoros is like Chananyah, and Shmuel rules like him. In Bava Metzi'a (95b), Abaye holds like R. Oshaya, and Rava holds like R. Yonason. We hold like Rava against Abaye, and we said that Chananyah holds like R. Yonason. However, even if the Halachah follows R. Yehudah, like it says in Kesuvos, he was unsure. Therefore, if we are sure which is the father, we do not slaughter the child and the father on the same day. A Mishnah (83a) discusses telling the buyer 'I sold the mother to be slaughtered today.' This discusses the usual case. People usually know the mother, but not the father. Bahag and the She'altos say that the Halachah does not follow Chananyah regarding Oso v'Es Beno. Presumably, what I said is correct, for we concluded that R. Yehudah was unsure.

Poskim

Shulchan Aruch (YD 16:2): Oso v'Es Beno applies to the mother. We know for sure which is its child. If we are sure which is the father, we do not slaughter the child and the father on the same day. If one slaughtered, he is not lashed, for it is a Safek whether it applies to males.

Beis Yosef (DH u'Mah): The Rashba says that the Ba'al ha'Ma'or and R. Tam are concerned for the father's seed, and he agreed. It is proper to be concerned for their opinion for a Torah Isur. Also the Rambam is concerned. Even though Bahag, the She'altos and the Rif rule like unlike Chananyah, the Rosh wrote that presumably we are concerned regarding Oso v'Es Beno. We hold like this.

Taz (2): The Gemara is unsure whether the Halachah follows Chananyah.

Gra (3): When we are unsure which animal was the father, we are not concerned regarding Oso v'Es Beno.

Gra (4): R. Yehudah, R. Eliezer and Rabanan are all unsure whether we are concerned for the father's seed.

Pleisi (1): It is a Safek whether it applies to the father. Perhaps one may slaughter the father Bein ha'Shemashos, for it is a Sefek-Sefeka. Even if it applies to the father, perhaps it is already night.

Rebuttal (Pischei Teshuvah 1, citing Zichron Yitzchak): This is Davar she'Yesh Lo Matirim (it is totally permitted tomorrow), therefore we do not permit due to Sefek-Sefeka.

Note: Also, Tosfos (Bava Kama 11a DH d'Ka) does not permit due to a Sefek-Sefeka if one may come to rely on contradictory leniencies. Perhaps at night he will slaughter another child of the father due to Sefek-Sefeka, for perhaps he slaughtered the father during the day!

Shulchan Aruch (8): If a female deer mated with a goat and a female was born, and one slaughtered the daughter and her son on the same day, he is lashed.

Shach (17): The Levush and Bach asked why the Rambam and Shulchan Aruch rule that we are definitely concerned for the father's seed. If not, he would not be lashed, for it is a Safek. Above (Sa'if 2), they say that even if we know the father, one is not lashed, for it is a Safek whether Oso v'Es Beno applies to males. They answered that above, it is a Safek whether Oso v'Es Beno applies to the father. Perhaps we expound that it applies only to the mother, for the child follows the mother. Here, surely we are concerned for the father's seed to say that the goat's child is a partial Seh.

Rebuttal (Shach): One cannot distinguish like this. The Gemara said that Shmuel rules like Chananyah, and this is consistent with what he taught that Chachamim argue with R. Yehudah and say that all mules are one species. (I.e. since he is concerned for the father's seed, he holds that Oso v'Es Beno applies to fathers. According to the Levush and Bach, perhaps the Drashah exempts fathers from Oso v'Es Beno! - PF) Also, what is the source to be unsure about the Drashah? Also, the Beis Yosef says that the Rambam is unsure whether we are concerned for the father's seed, like R. Yehudah. Also, the Rambam (Hilchos Shechitah 14:4) and Mechaber (28:2) obligate Kisuy ha'Dam without a Berachah for the child of a Behemah and Chayah. If we are surely concerned for the father's seed, one should bless! The Bach and Prishah (there) answered that a partial Seh suffices for Oso v'Es Beno, but perhaps a full Chayah is needed for Kisuy. This is unlike the Sugya, which explicitly says that a partial deer suffices! Also, the Rambam (Hilchos Bikurim 9:5) and Mechaber (61:8) obligate half-Matanos for the child of a male deer and a goat, for "Seh" includes a partial Seh. They exempt the child of a female deer and a goat. This shows that we are unsure, so the owner can say 'bring a proof that we are concerned for the father's seed, and then I will give half-Matanos.' The Rambam (Hilchos Kil'ayim 9:6) and Mechaber (297:9) forbid what is born from a donkey with what is born from a horse, for we are unsure. If we were Vadai concerned for the father's seed, they would be permitted! The Yam Shel Shlomo (5:2) says that the Halachah follows Chananyah, and Oso v'Es Beno applies to fathers. In 5:3, he says that we are not concerned for the father's seed, and permits to slaughter the father and child on the same day. This requires investigation.