
Section 4.2 Interval arithmetic over a linearly ordered set 97
If a multiplication exists in V , we also obtain for all A D Œa
1
, a
2
:
(F) A Œo, o ^ B Œo, o ) A
B D Œa
1
b
1
, a
2
b
2
,
(G) A Œo, o ^ B Œo, o ) A
B D Œa
2
b
2
, a
1
b
1
,
(H) A Œo, o ^ B Œo, o ) A
B D Œa
2
b
1
, a
1
b
2
,
(I) A Œo, o ^ B Œo, o ) A
B D Œa
1
b
2
, a
2
b
1
,
Proof. (a) Theorem 3.11 implies that fIV , IR, , g is a monotone upper screen vec-
toid of fP V , PRg, which is multiplicative if fV , R, g is. The inclusion-isotony is a
simple consequence of (OV5) in PV , of the monotonicity of the rounding
: PV !
IV , and of (RG). The proofs of the properties (OV1, 2, 3, 4) and (A)–( ...