Operations that preserve integrability, and truncated Riesz spaces

For any real number p is an element of [1, +infinity), we characterise the operations RI -> R that preserve p-integrability, i.e., the operations under which, for every measure mu, the set L-p(mu) is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind sigma-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that R generates this variety. From this, we exhibit a concrete model of the free Dedekind sigma-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind sigma-complete Riesz spaces with weak unit, R is proved to generate this variety, and a concrete model of the free Dedekind a-complete Riesz spaces with weak unit is exhibited.