Title: Faithful Semitoric Systems,
Abstract: This paper consists of two parts. The first provides a review of the basic
properties of integrable and almost-toric systems, with a particular emphasis
on the integral affine structure associated to an integrable system. The second
part introduces faithful semitoric systems, a generalization of semitoric
systems (introduced by Vu Ngoc and classified by Pelayo and Vu Ngoc) that
provides the language to develop surgeries on almost-toric systems in dimension
4. We prove that faithful semitoric systems are natural building blocks of
almost-toric systems. Moreover, we show that they enjoy many of the properties
that their (proper) semitoric counterparts do.