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arXiv · 2103.08170

Free Banach lattices generated by a lattice and projectivity

Antonio Avilés·Gonzalo Martínez-Cervantes·José David Rodríguez Abellán·Abraham Rueda Zoca
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Abstract

In this article we deal with the free Banach lattice $FBL\langle \mathbb{L} \rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle \mathbb{L} \rangle$ is projective then $\mathbb{L}$ has a maximum and a minimum. On the other hand, we show that if $\mathbb{L}$ has maximum and minimum then $FBL\langle \mathbb{L} \rangle$ is $2$-lattice isomorphic to a $C(K)$-space. As a consequence, $FBL\langle \mathbb{L} \rangle$ is projective if and only if it is lattice isomorphic to a $C(K)$-space with $K$ being an absolute neighborhood retract. As an application, we characterize those linearly ordered sets and Boolean algebras for which the corresponding free Banach lattice is projective.

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Antonio Avilés, Gonzalo Martínez-Cervantes, José David Rodríguez Abellán, Abraham Rueda Zoca. 2021-03-15. Free Banach lattices generated by a lattice and projectivity. https://arxiv.org/abs/2103.08170

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