Impossibility of defining the class of \(L_ 0\)-algebras by means of identities (Q1078563)

Let \(A=(A,\leq,+,0,1)\) be a commutative partially ordered monoid with 0 as the least and 1 as the greatest element, in which for each a, b there is a smallest c such that \(a\leq b+c\). This c is denoted by a-b. Let the equality \(1-(1-x)=x\) hold in A for all x. It is proved that the structures A do not form an equational class. This class is an algebraic equivalent of classical logic without contraction rule.