If three people eat together they are obligated to say zimun*.
If he ate from food which has not been tithed,
Or from the first tithe that had Terumah* separated,
Or the second tithe which had been sanctified and redeemed,
Or a servant that had eaten an olive's volume*, or a Cusi*,
He must say zimun with them.
(But) if he ate untithed produce,
Or the first tithe that had not had Terumah separated,
Or the second tithe which had been sanctified and not redeemed,
Or a servant that ate less than an olive's volume*, or an idol worshipper,
One cannot say zimun with them.
1. Zimun. A ritual in which the head of the household invites the other diners to bentch, or say Birkas HaMazon, the blessing after a meal
2. Terumah. Tithe
3. An olive's volume. The amount required to eat before being able to say Birkas HaMazon
4. Cusi. aka Cuthean, a sect of Jews who had converted from paganism and then renounced the Oral Torah.
5. A servant that ate less than an olive's volume. He must say a Bracha Acharonah, or a blessing after not eating an olive's volume of bread, and for that brachah no zimun is said