halacha - Mamzeirim and Slaves

If Reuben's sister, Hanna, is a slave and she and Reuben have sexual intercourse, are their children slaves or mamzeirim?

Acc. to the Mishna in Yevamot 4:12-13, the offspring of a forbidden union, the penalty for which is karet, is a mamzer. That is the opinion of Rabbi Shimon haTeimani in mishna 13 - contra the opinions of Rabbi Akiva and Rabbi Yehoshua - and is understood to be the opinion of the sages (contra Akiva) in mishna 12.

Acc. to the Mishna in Ketubot 3:12, the offspring of such a forbidden union - one in which the woman would have been permitted to another - is a mamzer. The offspring, however, of another forbidden union - one in which the woman could not have been permitted to another - follows the woman in status. The practical upshot of this mishna is that the child of a gentile woman or a slave girl (neither of whom can contract kiddushin with anybody) is either a gentile or a slave.

If there is a brother and a sister (Reuben and Hanna in my example) and she is a slave, they now belong to two of the different types of relationships delineated from one another in Ketubot. They are siblings, and the penalty for sexual intercourse is karet (Mishna, Keritot 1:1; cf: Leviticus 18:9, 29). That means, based on both Ketubot and Yevamot, their offspring are mamzeirim. And yet, being a slave, she cannot contract kiddushin with any man, which means that their offspring (based on Ketubot) are slaves.

Answer

You must mean a non-Jewish (actually, quasi-Jewish) slave, a "shifcha." (I.e. she was born non-Jewish, then underwent a part-conversion when she became a slave.) A born-Jewish, "ama ivriyah" goes free automatically upon reaching puberty, so that case is moot.

I don't know whether the partial conversion given to a shifcha already wipes out all existing relationships (as it does with full-blown conversion), but the point is moot. As the last Mishnah in Kiddushin Chapter 3 observes (and the Halacha follows Rabbi Tarfon's opinion):

רבי טרפון אומר:‏
יכולין הם ממזרים ליטהר.‏
כיצד?‏
ממזר שנשא שפחה, הוולד עבד;‏
שחררו, נמצא הבן בן חורין.‏

The offspring of a male mamzer and female shifcha is just plain eved (eved kna'ani, to be specific), not eved mamzer. There is no room in the category for any flavor of eved, just plain eved. Thus to the best of my knowledge the same would apply here, the offspring of a shifcha is just plain eved; whether the father was a normal Jew, a non-Jew, a mamzer Jew, or a forbidden-relation Jew.