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In this paper we will introduce a certain type of morphisms of log schemes(in the sense of Fontaine, Illusie, and Kato) and investigate their moduli.Then by applying this we define a notion of toric algebraic stacks overarbitrary schemes, which may be regarded as torus embeddings within theframework of algebraic stacks, and study some basic properties. Furthermore, westudy the stack-theoretic analogue of toroidal embeddings.

Logarithmic geometry, minimal free resolutions and toric algebraic stacks