
54 Chapter 2 Ringoids and ve c toids
Theorem 2.7. If fR, C, g is a ringoid with the neutral elements o and e,then
fCR, C, g is also a ringoid with the neutral elements ! D .o, o/ and D .e, o/.
Moreover D .e, o/.IffR, N , C, , =g is a division ringoid, then fCR,
N , C, , =g
is also a division ringoid. If fR, C, , g is a weakly ordered ringoid, fCR, C, , g is
likewise a weakly ordered ringoid.
Proof. The properties (D1, 2, 3, 4, 7, 8) and (OD1, 2) follow directly from the defini-
tion of the operations and the corresponding properties in R. The properties (D5) and
(D9) can be shown to be valid for D .e, o/ 2 CR: (D5a) and (D5b) are easily ...