Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.

In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.

The survival distribution of a device subject to a sequence of shocks occurring randomly over time is studied by Esary, Marshall and Proschan (1973) and by A-Hameed and Proschan (1973), (1975). The present note treats the case in which shocks occur according to a homogeneous Poisson cluster process. It is shown that if [the device survives k shocks] = zk, 0 < z < 1, then the device exhibits a decreasing failure rate. A DFR preservation theorem is proved for completely monotonic . A counterexample to the IFR preservation theorem is given in which is strictly IFR while the failure rate is initially decreasing and then increasing.