Intervals in l-groups as L-algebras

L-algebras are related to algebraic logic and quantum structures. They were introduced by the first author [J. Algebra 320 (2008)], where a self-similar closure S(X) of any L-algebra X was employed to derive a criterion for X to be representable as an interval in a lattice-ordered group. In the present paper, this criterion is improved without using the embedding. It is shown that an L-algebra is representable as an interval in a lattice-ordered group if and only if it is semiregular with a smallest element and bijective negation. Any such L-algebra gives rise to a perfect dual with respect to the inverse of the negation. This is proved by a self-dual characterization of semiregularity.