Cignoli, Roberto; Torrens, Antoni Free cancellative hoops. (English) Zbl 1012.06015 Algebra Univers. 43, No. 2-3, 213-216 (2000). 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. Reviewer: Jiří Rachůnek (Olomouc) Cited in 10 Documents MSC: 06F05 Ordered semigroups and monoids 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 08B20 Free algebras Keywords:cancellative hoop; ordered monoid; abelian \(l\)-group; free algebra PDFBibTeX XMLCite \textit{R. Cignoli} and \textit{A. Torrens}, Algebra Univers. 43, No. 2--3, 213--216 (2000; Zbl 1012.06015) Full Text: DOI