Arithmetic operations for fuzzy numbers induced by Gaussian kernels

In this paper we consider fuzzy numbers which is uniquely identified by a class of normalized membership functions. The function space, denoted by
$X_{h, G}$
, is constructed by combining a sigmoid function
$h$
(subjective perception) and a class of Gaussian kernel functions
$G(\cdot,\mu^{-},\mu^{+})$
(objective entity), respectively. Especially, By means of the free variables
$(\mu^{-}, \mu^{+})$
, we define the addition, scalar multiplication and subtraction on
$X_{h, G}$
. We claim that, under our definitions,
$X_{h, G}$
has a linear algebra.