We deal with first-order definability in the substructure ordering ( 𝒟 ; ⊑ ) $({\mathscr{D}}; \sqsubseteq )$ of finite directed graphs. In two papers, the author has already investigated the first-order language of the embeddability ordering ( 𝒟 ; ≤ ) $({\mathscr{D}}; \leq )$ . The latter has turned out to be quite strong, e.g., it has been shown that, modulo edge-reversing (on the whole graphs), it can express the full second-order language of directed graphs. Now we show that, with finitely many directed graphs added as constants, the first order language of ( 𝒟 ; ⊑ ) $({\mathscr{D}}; \sqsubseteq )$ can express that of ( 𝒟 ; ≤ ) $({\mathscr{D}}; \leq )$ . The limits of the expressive power of such languages are intimately related to the automorphism groups of the orderings. Previously, analogue investigations have found the concerning automorphism groups to be quite trivial, e.g., the automorphism group of ( 𝒟 ; ≤ ) $({\mathscr{D}}; \leq )$ is isomorphic to ℤ 2 $\mathbb {Z}_{2}$ . Here, unprecedentedly, this is not the case. Even though we conjecture that the automorphism group is isomorphic to ( ℤ 2 4 × S 4 ) ⋊ α ℤ 2 $(\mathbb {Z}_{2}^{4} \times S_{4})\rtimes _{\alpha } \mathbb {Z}_{2}$ , with a particular α in the semidirect product, we only prove it is finite.

中文翻译：

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有限有向图子结构排序的可定义性

我们处理有限有向图的子结构排序 ( 𝒟 ; ⊑ ) $({\mathscr{D}}; \sqsubseteq )$ 中的一阶可定义性。在两篇论文中，作者已经研究了可嵌入排序的一阶语言 ( 𝒟 ; ≤ ) $({\mathscr{D}}; \leq )$ 。后者已经证明是非常强大的，例如，已经表明，模边反转（在整个图上），它可以表达有向图的完整二阶语言。现在我们证明，将有限多个有向图作为常量添加， ( 𝒟 ; ⊑ ) $({\mathscr{D}}; \sqsubseteq )$ 的一阶语言可以表达 ( 𝒟 ; ≤ ) $({ \mathscr{D}}; \leq )$ 。这种语言表达能力的限制与排序的自同构群密切相关。之前，类似的研究发现有关的自同构群非常简单，例如， ( 𝒟 ; ≤ ) $({\mathscr{D}}; \leq )$ 的自同构群与 ℤ 2 $\mathbb {Z}_ {2}$。在这里，史无前例的情况并非如此。尽管我们推测自同构群同构于 ( ℤ 2 4 × S 4 ) ⋊ α ℤ 2 $(\mathbb {Z}_{2}^{4} \times S_{4})\rtimes _{\ alpha } \mathbb {Z}_{2}$ ，在半直积中有一个特定的 α，我们只证明它是有限的。