# Free sequences in \({\mathscr {P}}\left( \omega \right) /\text {fin}\)

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## Abstract

We investigate maximal free sequences in the Boolean algebra \({\mathscr {P}}\left( \omega \right) {/}\text {fin}\), as defined by Monk (Comment Math Univ Carol 52(4):593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \({\mathfrak {f}}\). Answering a question of Monk, we demonstrate the consistency of \(\omega _1 = {\mathfrak {i}} = {\mathfrak {f}} < {\mathfrak {u}} = \omega _2\). In fact, this consistency is demonstrated in the model of Shelah for \({\mathfrak {i}}< {\mathfrak {u}}\) (Shelah in Arch Math Log 31(6):433–443, 1992). Our paper provides a streamlined and mostly self contained presentation of this construction.

## Keywords

Maximal free sequence Dense independent system Party forcing## Mathematics Subject Classification

03E17 03E35 06E05## Notes

### Acknowledgements

The authors would like to thank Osvaldo Guzmán for numerous suggestions substantially improving the paper.

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