Gemara

27a (Mishnah): The four species (altogether) taken with the Lulav are Me'akev each other.

It says about the four species including the Lulav - "ul'Kachtem". The Kichah (taking) must be Tamah (complete);

(Rav Chanan bar Rava): This is only if all four species are not in front of him. If they are in front of him, they are not Me'akev.

Question (Beraisa): Two of the four species (Esrog and Lulav) come from trees that bear fruit, and two do not. Each pair requires the other. To fulfill the Mitzvah they must be bound together (except for the Esrog);

Similarly, Hash-m does not accept the prayers of Yisrael until they are united - "ha'Boneh va'Shamayim Ma'alosav va'Agudaso Al Eretz Yesadah".

Answer: Tana'im argue about this:

(Beraisa): One fulfills the Mitzvah of the four species even if they are not tied together;

R. Yehudah says, if they are tied together they are Kesherim. If not, they are Pesulim.

Rishonim

Rif (Sukah 17a): Rav Chana taught that if he has all four, they are not Me'akev. Even if he took each one by itself, he was Yotzei. We hold that the Lulav need not be tied.

Rosh (Sukah 3:14): Also Bahag explains like this. R. Tam disagrees. Since it is all one Mitzvah, it is unreasonable that it helps to take one after the other. He emended the text to say 'they are Me'akev each other due to Tamah only when he does not have them. If he has them, this is Tamah, even if he did not tie them in a bundle.' This is when all four species are not in front of him. If they are in front of him, they are not Me'akev. This is like Rabanan, who do not expound "Lekichah-Lekichah."

Rebuttal (Rosh): R. Tam's text and Perush are unreasonable. If Rav Chanan merely comes to teach that the Halachah follows Rabanan, that the Lulav need not be tied, he should say 'the Halachah follows Rabanan'! Also, he need not teach this, for the Halachah follows the Rabim! Rather, he teaches that if they are in front of him, he blesses and takes one at a time. Presumably, he takes the Lulav first, for the Berachah mentions it.

Ran (DH v'Af): The Gemara connotes unlike R. Tam. They say that the Ramban was even more lenient. Even if they were not all in his Reshus together, as long as he took all of them, he was Yotzei. We say that when he does not have all of them they are Me'akev, i.e. if he did not take all four he was not Yotzei. However, if he took all of them, even one after the other, he was Yotzei. Even so, it is improper to bless on the Lulav unless he is sure that he will get all of them. Even though we bless only on the Lulav, since if he will not get all of them he was not Yotzei, perhaps he will not get all of them and his Berachah was l'Vatalah.

Beis Yosef (OC 651 DH v'Gam): The Rambam disagrees with the Ramban. He says not to take them until all four are in front of him.

Rambam (Hilchos Lulav 7:5): The four Minim are one Mitzvah. They are Me'akev each other.

Rambam (6): The ideal Mitzvah is to tie the Lulav, Hadasim and Aravos in one bundle. If he did not tie them, and took them one by one, he was Yotzei. This is if all were available to him. If he had only one Min, or he was missing one of the Minim, he does not take until he finds the rest.

Hagahos Maimoniyos (7): R. Simchah says that if he talked in between one Min and another, he blesses again.

Poskim

Shulchan Aruch (OC 651:12): If all four Minim are available to him, if he took them one at a time, he was Yotzei.

Beis Yosef (DH v'Gam): The Rif, Rambam, Rosh, Rashi and Ran hold unlike R. Tam.

Bach (DH u'Mah): This connotes that one need not touch the Esrog to the Lulav at the time of the Berachah. One is Yotzei even if he takes them one after the other, if they were in front of him! The Rikanti says that based on Kabalah, one must put them together. Surely, this is just for the ideal Mitzvah, but it is not Me'akev.

Rema: This is only if all were in front of him. He takes the Lulav and blesses on it, with intent for the others.

Beis Yosef (DH v'Zeh, citing Orchos Chayim): The Ra'avad says that if he took each by itself, since they are not tied together, the Berachah on the Lulav does not exempt the others. He blesses on each by itself. In practice, I say to bless only on the Lulav, like the Rambam.

Beis Yosef (DH v'Chosav Od, citing Orchos Chayim): R. Shalmi (Hagahos Tur ha'Shalem 32 - it seems that this should say 'R. Shlomo' or 'R. Shalmiyah') says that if one took both (the Esrog and the Lulav bundle) in one hand, he was not Yotzei, for it says "Kapos", and not 'v'Kapos'.

Mo'adim u'Zmanim (2:121): Some say that one is not Yotzei if they are in one hand, so one must be stringent for the Torah Mitzvah (when blessing on Yom Tov), but one may be lenient at other times

Magen Avraham (24): They should all be in front of him so he will not need to interrupt between them.

Mishnah Berurah (5): Even if he knows that he will get the other Minim, he may not bless until all are in front of him.

Kaf ha'Chayim (118): If one takes them one at a time, he takes also the Esrog in his right hand.

Rema (ibid.): If he talked in between, he must bless on each one by itself.

Beis Yosef (DH Kasuv b'Orchos): A case occurred in which R. Meshulam blessed Al Netilas Lulav and Shehecheyanu on the first day, and afterwards saw that there is no Aravah. He put an Aravah in with the Lulav, and blessed Al Netilas Aravah and Shehecheyanu, due to the Aravah. He could have taken it by itself; it was Hidur Mitzvah to take all the four Minim together. Also the Ra'avad says so.

Magen Avraham (25): Hagahos Maimoniyos cites R. Simchah, who says that if he talked in between the Minim, he blesses again. This is like R. Simchah says about one who taked in between Teki'os. We rule that it is all one Mitzvah, like it says in Menachos, so he does not bless again. Perhaps the Rema says to bless again because the Ra'avad holds that one blesses on each even if he did not talk in between. The Beis Yosef (672) cited Orchos Chayim to say that if one blessed on Ner Chanukah on the fourth night, and saw that there were only three Neros, he does not bless again, for his first Berachah was for all the Neros he must light. If one blessed on bread (or Peros or wine), and later they brought to him an accompaniment (or more Peros or wine) that was not in front of him when he blessed, we say that he intended for it. Orchos Chayim and the Ra'avad hold that one must bless on each, but we rule unlike them, unless one talked in between. This is why the Mechaber omitted this.

Eshel Avraham (25): The Rema (592:3) rules that one who talked in between the Teki'os of Shacharis and of Musaf does not bless again, for it is all one Mitzvah, unlike Tefilin. Here the Rema rules like R. Simchah that he blesses again, even though it is one Mitzvah! The Magen Avraham answered that here he blesses due to a Sefek-Sefeka (two doubts). Perhaps the Halachah follows the Ra'avad (and one blesses on each even if he did not talk), and even if not, perhaps the Halachah follows R. Simchah.

Machatzis ha'Shekel (DH v'Im): The Magen Avraham compares this to blessing on bread. Just like one who blessed on bread and ate immediately may talk during the meal, if one blessed on a Mitzvah and began the Mitzvah, if he talked afterwards it is not a Hefsek (interruption). Tefilin is different, for it is two Mitzvos.

Mishnah Berurah (56): We learn from here that if one blessed on the Lulav, and afterwards noticed that the Hadas or Aravah was missing or Pasul or upside down, if he had a Kosher one in his house and he would not need to interrupt to get it, he takes all four and blesses on the Min he was missing, i.e. 'Al Netilas Aravah' or 'Al Netilas Etz Avos.'. If he did not have a Kosher when in his house when he blessed, and later they gave to him, he blesses again 'Al Netilas Lulav.' Since they were not available to him, he was not Yotzei when he blessed the first time. In either case, if it was the first day, he blesses Shehecheyanu again.

Gra (DH v'Daito): This is like we say (36a) that if one talked in between Tefilin Shel Yad and Tefilin Shel Rosh, he blesses again, each one according to his opinion (about whether or not one blesses on Tefilin Shel Rosh if he did not talk in between).

Kaf ha'Chayim (122): Since it is a Safek, he should bless again in his heart. This way there is no concern for a Hefsek or a Berachah l'Vatalah.

Bi'ur Halachah (DH v'Im): I say that Hagahos Maimoniyos holds that even though we do not hold like R. Tam, who requires taking all of them at once, we require one after the other without a Hefsek in between. If there is a Hefsek, surely this is not Lekichah Tamah. However, if so he was not Yotzei even for the Lulav, and he should bless again Al Netilas Lulav! The Rema connotes unlike this. Hagahos Maimoniyos says that if he talked in between, he blesses again. Perhaps he means that he blesses again Al Netilas Lulav. Perhaps the Rosh, Tur and R. Yerucham, who say that if he took them one after the other he was Yotzei, require that he not talk needlessly in between. The Birkei Yosef brought that Hagahos Maimoniyos is unlike the Ritva. This is not difficult. The Ritva was a Talmid of the Ramban, and we rule unlike the Ramban.