Logarithmic Structure for Stable Maps Relative to Simple Normal Crossing Divisor

P gp := {(a, b)|(a, b) ∼ (c, d) if ∃s ∈ P such that s+ a+ d = s+ b+ c}. The monoid P is called integral if the natural map P → P gp is injective. And it is called saturated if it is integral and satisfies that for any p ∈ P , if n · p ∈ P for some positive integer n then p ∈ P . A monoid P is said to be fine if it is integral and finitely generated. A monoid P is called sharp if there are no other unit except 0. A nonzero element p in a sharp monoid P is called irreducible if p = a+ b implies either a = 0 or b = 0. We denote by Irr(P ) the set of irreducible elements in a sharp monoid P . A fine monoid P is called free if P ∼= N for some positive integer n. A monoid P is called torsion free if the associated group P gp is torsion free. The monoid P is called toric if P is fine, saturated, and torsion free. This is the monoid we will use in this paper. A morphism h : Q → P between integral monoids is called integral if for any a1, a2 ∈ Q, and b1, b2 ∈ P which satisfy h(a1)b1 = h(a2)b2, there exist a2, a4 ∈ Q and b ∈ P such that b1 = h(a3)b and a1a3 = a2a4. ss:DefLogStr 1.1.2. Logarithmic structures. Let X be a scheme. A pre-log structure on X is a pair (M, exp), which consists of a sheaf of monoidsM on the étale site Xét of X, and a morphism of sheaves of monoids exp :M→ OX , called the structure morphism of M. Here we view OX as a monoid under multiplication.