Some basic properties of generalized Calderon-Lozanovski\v{\i} spaces

Generalized Calderon-Lozanovski\v{\i} spaces $\Psi_\varphi(E_1,E_2)$ introduced in [3] are investigated. These spaces are generated by a function $\Psi : T\times R^2\rightarrow R_+$ such that $\Psi(\cdotp, u)$ is a $\Sigma$-measurable function for any $u\in R^2$ and $\Psi(t,\cdotp)$ is a homogeneous, concave function vanishing at zero and by a couple of Banach function lattices $E_1$ and $E_2$ over a nonatomic measure space $(T,\Sigma,\mu)$. We investigate the special class of these spaces, namely the spaces $E_\varphi$ corresponding to $\Psi(E,L^\infty)$, where $E$ is an arbitrary Banach function lattice. We investigate the problem of order continuity, Fatou property, property $\textbf{H}_\mu$ and order isomorphically isometric copies of $l^\infty$ in $E_\varphi$. We also consider some relationships between the norm and the modular as well as between the modular convergence and the norm convergence in $E_\varphi$. In order to do so, we define a regularity condition $\Delta^E_2$