A log scheme is a scheme \(X\) endowed with an étale sheaf of monoids \({\mathcal M}_X\) and with a map of monoids \(\alpha:{\mathcal M}_X\to{\mathcal O}_X\) (\({\mathcal M}_X\) is regarded as a multiplicative sheaf of monoids) such that the induced map \(\alpha^{-1}({\mathcal O}^*_X)\to{\mathcal O}_X^*\) is an isomorphism. The author associates to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, called the associated graded algebra of the given log scheme. Then the author constructs a canonical connection on this algebra. In the log smooth case, over a field \(k\) of characteristic \(p\) the author studies the integrable and \(p\)-integrable graded modules over this algebras and proves a Cartier type \(p\)-descent theorem, which generalizes a result of \textit{A. Ogus} [``\(F\)-crystals, Griffiths transversality, and the Hodge decomposition'', Astérisque 221, Société Mathématique de France (1994; Zbl 0801.14004)].