Local structure of generalized Orlicz−Lorentz function spaces

We study the local structure of a separated point $x$ in the generalized Orlicz-Lorentz space ${\mathrm{\Lambda }}^{\phi }$ which is a symmetrization of the respective Musielak-Orlicz space $L^{\phi }$. We present criteria for an $LM$ point and a $\mathit{UM}$ point, and sufficient conditions for a point of order continuity and an $\mathit{LLUM}$ point, in the space ${\mathrm{\Lambda }}^{\phi }$. We prove also a characterization of strict monotonicity of the space ${\mathrm{\Lambda }}^{\phi }$.