On categorical equivalences of commutative BCK-algebras (Q1970590)

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra \((X;*,0)\) which satisfies the condition: if \(a\leq x\), \(a\leq y\) and \(x*a=y*a\), then \(x=y\). Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian \(l\)-groups whose objects are pairs \((G,G_0)\), where \(G\) is an Abelian group, \(G_0\) is a subset of the positive cone generating \(G^+\) such that if \(u\), \(v\in G_0\), then \(0\vee(u-v)\in G_0'\), and morphisms are \(l\)-group homomorphisms \(h:(G,G_0)\to(G',G_0')\) with \(f(G_0)\subset G_0'\).