We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it. We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the relative expressive power of ordered logics, and to characterise our logics as bisimulation-invariant fragments of FO a la van Benthem. Afterwards, we study the Craig Interpolation Property~(CIP). We refute yet another claim from the infamous work by Purdy, by showing that the fluted and forward fragments do not enjoy CIP. We complement this result by showing that the ordered fragment and the guarded ordered logics enjoy CIP. These positive results rely on novel and quite intricate model constructions, which take full advantage of the "forwardness" of our logics.

翻译：我们认为,这些逻辑的复杂程度现已确立,但其模型理论却鲜为人知。我们的论文旨在对此提供一些洞察力。我们首先为定购逻辑提供适当的平衡概念。我们接下来采用平衡法,比较定购逻辑的相对表达力,并将我们的逻辑定性为范本特姆FO的振奋-不易动的碎片。随后,我们研究Craig Indigation Institute Property~(CIP),我们反驳了Purdy的恶名工作提出的另一项主张,表明流出和前方的碎片并不享受CIP。我们通过表明定购的碎片和定购逻辑享有CIP来补充这一结果。这些积极的结果依赖于新颖和相当复杂的模型结构,这些模型充分利用了我们“反向”逻辑的优势。