Residuated lattices and lattice effect algebras (Q879018)

The authors study two partial operations in effect algebras, namely \(a \odot b = (a' \oplus b')'\) if \(a' \oplus b'\) is defined and \(a \to_p b = a' \oplus b\) if \(a' \oplus b\) is defined. They show, e.g., that for an effect algebra \((E, \oplus, 0, 1)\) we obtain a (dual) effect algebra \((E, \odot,0,1)\). Using these operations they construct an adjoint pair in an effect algebra with the Riesz decomposition property to obtain an involutive residuated lattice. On the other hand, they prove that an involutive residuated lattice \((L, \leq \otimes, \to, 0, 1)\) corresponds to an effect algebra with the Riesz decomposition property if and only if \(a \land b = a \otimes (a \to b)\) for every \(a,b \in L\).