A half factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on \mathbb {N}$N*
The half factorial, locally right Garside monoid of cofinite monotone sequences on \mathbb {N}$N* is considered in this paper. This is a right cancellative, left rigid Mobius monoid, and this monoid directs us to the inverse monoid of cofinite monotone partial bijections on \mathbb {N}$N* In a certain sense, the polycyclic inverse monoid is a 0 analogue of this derived inverse monoid. Similarities and differences between these two inverse monoids are presented involving Mobius categories, Leech's construction, breaking processes, gauge inverse submonoids, etc.