A conical space is a set equipped with a notion of taking real-linear combinations with nonnegative coefficients of its elements. We may tersely define it as a module of the rig : that is, the rig of nonnegative real numbers, with ordinary addition and multiplication as the rig operations.
Any rig homomorphism gives a ‘restriction of scalars’ functor and ‘extension of scalars’ functor . In particular, the rig homomorphism
In simple English: any real vector space has an underlying conical space, and any conical space freely generates a real vector space.
An extended conical space is a module over the rig of nonnegative extended real numbers, with . For purposes of constructive mathematics, one should take to be the space of nonnegative lower real numbers.
Since we have a rig homomorphism , every extended conical space has an underlying conical space, and any conical space freely generates an extended conical space. However, there is no direct relationship between vector spaces and extended conical spaces.