Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of \emph{additive} classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder.

翻译：逻辑转换提供了一种非常有用的工具, 用于将其他结构类别内部的结构分类编码。 在本文中, 我们研究的是第一阶( FO) 转换及其在无限的限定图表类别上产生的准顺序。 令人惊讶的是, 这一准顺序非常复杂, 虽然是由一阶逻辑的局部特性所决定的。 这与假设的月经第二阶( MSO) 转换准顺序的简单性形成对照。 我们首先为FO 转换建立本地普通格式, 这是独立感兴趣的。 然后我们证明, 商序部分顺序是约束性分配组合组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合组合组合组合式组合式组合式组合组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合组合组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合组合组合组合组合组合组合组合组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合组合组合组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合式组合