Commentaire sur Téroumot 3:4

במה דברים אמורים – that which Rabbi disputes above and states that the heave-offering of both is Terumah/heave-offering.

To what does this apply? Only if one did not confer with the other, but if one allows a member of his household, or his slave or female slave to give terumah for him, this terumah is terumah.
If he annulled [this permission]: If he annulled it before the taking of the terumah, the terumah is not terumah; But if he annulled it after the terumah had been taken, the terumah is terumah.
Workers have no permission to give terumah, except for those who tread [grapes] for they defile the winepress immediately.
Today’s mishnah continues the discussion in yesterday’s mishnah concerning a partner who gives terumah from shared property. Our mishnah teaches that one person can give another member of his household permission to give terumah on his behalf.
Section one: This section is difficult for the mishnah begins by discussing partners but backs this up by mentioning members of one’s household. It seems that what the mishnah intends to say is that one partner agreed to that which the first person gave, then the first terumah is terumah. Albeck interprets this to be a continuation of Rabbi Yose’s words. Rabbi Yose says that the second terumah is valid if the first person did not give the proper measure of terumah. This is a case where the second partner did not agree with what the first partner having given less than the proper measure. However, if the second person did agree with what the first partner gave, then the terumah is valid despite its not being the proper measure because the Torah does not prescribe a fixed amount for terumah. The measure of terumah was only set by the sages.
Along with this, the mishnah teaches that a householder can give the members of his household permission to separate terumot on his behalf. Household members would include the slaves.
Section two: He may annul the permission that he gives to the members of his household to separate terumah. However, he must annul this permission before they separate the terumah. If he does so after they have already set aside the terumah, then their terumah is still valid.

בשלא דבר – that this one did not receive permission from his colleague to separate the heave-offering, but rather, he separated the heave-offering of his own accord.

אבל הרשה – that the owner gave permission [to his colleague] to separate the heave-offering, his heave-offering is a heave-offering.

בטל – if afer he had appointed an agent to separate the heave-offering, he cancelled his agency after the agent went from before him.

חוץ – the owners who are illiterate who hired workers who are members of the order for the observance of Levitical laws in daily intercourse in order to separate the heave-offering in ritual purity, are permitted to separate the heave offering until the illiterate owners arrived who would defile the heave-offering, because they ritually defile the vat for wine pressing, for the illiterate owners immediately when the when those who tread began to stamp and they went to the vat for wine pressing in the longitudinal and latitudinal directions are not more careful from coming into contact in their vas for wine-pressing and defile it, because they think that immediately when they begin to tread, their colleagues had separated out the Great Terumah/heave-offering (i.e., two percent for the Kohen). Therefore, they separate the members of the order for the observance of Levitical laws in daily intercourse that stamp from the wine in order [to reach] the measure of heave-offering that is necessary for the vat for wine-pressing immediately when they walked in it longitudinally and latitudinally. If the owners would come and defile the vat for wine pressing prior to their completing it, they have designated wine to separate everything in ritual purity.