Copies of $$l^{\infty }$$ in Quasi-normed Calderón–Lozanovskiĭ Spaces

Abstract We consider order isomorphic and order linearly isometric copies of $$l^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> in the quasi-normed Calderón–Lozanovskiĭ spaces $$E_{\varphi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>φ</mml:mi> </mml:msub> </mml:math> . We present a number of theorems describing these copies in the natural language of suitable properties of the quasi-normed ideal space E and the non-decreasing Orlicz function $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> . In particular, we will characterize quasi-normed Orlicz–Lorentz spaces having order isomorphic and order linearly isometric copies of $$l^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> . Our studies are conducted in a full possible generality due to the measure space and the Orlicz function.