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On the order structure of Orlicz lattices. (English) Zbl 0767.46005
Summary: We investigate the order structure of \(\delta\)-Dedekind complete Riesz spaces \(X\) on which some functional \(\rho\) called a modular is defined. The pairs \((X,\rho)\) will be called Orlicz lattices. Some additional conditions relating properties of \(\rho\) to the order structure of \(X\) are considered. Under these conditions imposed on \(\rho\), we give some characterizations of sequential order convergence, sequential order star- convergence and order boundedness in \(X\) in terms of the modular \(\rho\).

46A80 Modular spaces
46A40 Ordered topological linear spaces, vector lattices
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)