On (weak) zero-fixing isometries in dually residuated lattice ordered semigroups (Q2777500)

An algebra \({\mathbf A}=(A,+,-,0,\vee ,\wedge)\) of type \((2,2,0,2,2)\) is called a dually residuated lattice-ordered semigroup if \((A,+,-,0)\) is an abelian monoid, \((A,\vee ,\wedge)\) is a lattice and if for all \(x,y,z\in A\) there holds \((x\vee y)+z=(x+z)\vee(y+z)\), \((x-y)+y\geq x\) and \(((x-y)\vee 0)+y\leq x\vee y\), and if \(z+y\geq x\) implies \(z\geq x-y\). A mapping from \(A\) onto \(A\) is called a zero-fixing isometry of \({\mathbf A}\) if \(f(0)=0\) and \(\rho(f(x),f(y))=\rho(x,y)\) for all \(x,y\in A\) where \(\rho(x,y):=(x-y)\vee(y-x)\) for all \(x,y\in A\). It is proved that the group of zero-fixing isometries of a dually residuated lattice-ordered semigroup is isomorphic to the group of zero-fixing isometries of an abelian lattice-ordered group.