A.T. Robertson Commentary 1 Corinthians 10:18

"Behold Israel after the flesh: have not they that eat the sacrifices communion with the altar?" — 1 Corinthians 10:18 (ASV)

After the flesh (κατα σαρκα). The literal Israel, the Jewish people, not the spiritual Israel (Ισραηλ κατα πνευμα) composed of both Jews and Gentiles, the true children of faith (Romans 2:28; Romans 9:8; Galatians 3:7).

Communion with the altar (κοινωνο του θυσιαστηριου). Same idea in κοινωνο participators in, partners in, sharers in (with objective genitive). The word θυσιαστηριον is from late verb θυσιαζω, to offer sacrifice, and that from θυσια, sacrifice, and that from θυω, common verb to slay, to sacrifice (verse 20). The Israelites who offer sacrifices have a spiritual participation in the altar.