The enigma of ’’entropy’’ is reconsidered from the viewpoint of generalized information theory on a lattice generated from measurements that define the system. A small (incomplete) set of natural axioms for a global information measure is developed sufficiently to deduce as a special case a generalization of Segal’s entropy on a W*-algebra (classical and quantum). A simple relationship between monotonicity of entropy and a semigroup on [0,∞] (representing composibility of information) is presented. Various extensions of information-theoretic results are incidentally proven, including relations between regular composible informations (on an orthocomplemented complete lattice) and measures (on σ—ideals of the lattice).