Free cancellative hoops (Q1866810)

A cancellative hoop is an algebra \(\langle A;+,\ominus ,0\rangle\) of type \(\langle 2,2,0\rangle\) such that \(\langle A;+,0\rangle\) is a commutative monoid and the following axioms are satisfied: \(x+(y \ominus x)=y+(x \ominus y)\); \((x \ominus y) \ominus z=x \ominus (y+z)\); \(x \ominus x=0\); \(0 \ominus x=0\); \((x+y) \ominus x=y\). It is known that the positive cone \(G^+ \) of any abelian \(l\)-group \(G\) can be considered as a cancellative hoop, and, conversely, an algebra \(A\) of type \(\langle 2,2,0\rangle\) is a cancellative hoop if and only if there is an abelian \(l\)-group \(G\) such that \(A\) is isomorphic to \(G^+ \). The authors of the paper describe the free cancellative hoops in terms of piecewise linear functions.