
Section 2.2 Vectoids 59
Once again let us consider matrices over a given ringoid R. In addition to the inner
operations and the order relation defined above in M
mn
R, we now define an outer
multiplication by
^
a2R
^
.a
ij
/2M
mn
R
a .a
ij
/ :D .a a
ij
/.
Then the following theorem holds.
Theorem 2 .12. Let fR, C, g be a ringoid with the neutral elements o and e.Then
fM
mn
R, Rg is a vectoid. The neutral element is the matrix all components of
which are o.IffR, C, , g is a weakly ordered (resp. an ordered) ringoid, then
fM
mn
R, R, g is a weakly ordered (resp. an ordered) vectoid.
Proof. We omit the proof of this theorem since it is straightforward.
For m D 1 we