 There are logics for which an algebraic semantics is provided in the literature in a slightly different way from the one given by the schema above. Let us consider the example in  Section 3.5  of Intuitionistic Linear Logic without exponentials. We denote by \(\bILsubZ\) the class of IL-algebras with zero defined in Troelstra 1992 (but adapted to the language of \(\bILL)\). Each \(\bA \in \bILsubZ\) is a lattice with extra operations and thus has its lattice order \(\le^{\bA}\). This lattice order has a greatest element which we take as the interpretation of \(\top\). On each one of these algebras \(\bA\) there is a designated element \(1^{\bA}\) (the interpretation of the constant 1) that may be different from the greatest element. It holds: