Vines and partial transformations

We introduce the partial vine monoid PVn. This monoid is related to the partial transformation semigroup PTn in the same way as the braid group Bn is related to the symmetric group Sn, and contains both the vine monoid [T.G. Lavers, The theory of vines, Comm. Algebra 25 (4) (1997) 1257–1284] and the inverse braid monoid [D. Easdown, T.G. Lavers, The inverse braid monoid, Adv. Math. 186 (2) (2004) 438–455]. We give a presentation for PVn in terms of generators and relations, as well as a faithful representation in a monoid of endomorphisms of a free group. We also derive a new presentation for PTn.