We study the correspondence theory of intuitionistic modal logic in modal Fairtlough–Mendler semantics (modal FM semantics) (Fairtlough and Mendler in Inf Comput 137(1):1–33, 1997), which is the intuitionistic modal version of possibility semantics (Holliday in UC Berkeley working paper in logic and the methodology of science, 2022. http://escholarship.org/uc/item/881757qn). We identify the fragment of inductive formulas (Goranko and Vakarelov in Ann Pure Appl Logic 141(1–2):180–217, 2006) in this language and give the algorithm \(\textsf{ALBA}\) (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012) in this semantic setting. There are two major features in the paper: one is that in the expanded modal language, the nominal variables, which are interpreted as atoms in perfect Boolean algebras, complete join-prime elements in perfect distributive lattices and complete join-irreducible elements in perfect lattices, are interpreted as the refined regular open closures of singletons in the present setting, similar to the possibility semantics for classical normal modal logic (Zhao in J Logic Comput 31(2):523–572, 2021); the other feature is that we do not use conominals or diamond, which restricts the fragment of inductive formulas significantly. We prove the soundness of \(\textsf{ALBA}\) with respect to modal FM-frames and show that \(\textsf{ALBA}\) succeeds on inductive formulas, similar to existing settings like (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012; Zhao 2021, in: Cia-battoni, Pimentel, Queiroz (eds) Logic, language, information, and computation, Springer International Publishing, Cham, 2022).

我们研究了模态 Fairtlough-Mendler 语义（模态 FM 语义）中直觉模态逻辑的对应论（Fairtlough and Mendler in Inf Comput 137(1):1–33, 1997），这是可能性语义的直觉模态版本（Holliday加州大学伯克利分校逻辑与科学方法论工作论文，2022 年。http://escholarship.org/uc/item/881757qn）。我们用这种语言识别归纳公式的片段（Goranko 和 Vakarelov in Ann Pure Appl Logic 141(1–2):180–217, 2006）并给出算法 \(\ textsf{ALBA}\)（Conradie 和 Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012）在这种语义设置中。论文有两个主要特点：一是在扩展模态语言中，名义变量被解释为完美布尔代数中的原子、完美分配格中的完全联素元和完美格中的完全联不可约元。 ，被解释为当前环境中单例的精炼正则开闭包，类似于经典正则模态逻辑的可能性语义（Zhao in J Logic Comput 31(2):523–572, 2021）；另一个特点是我们不使用共项或菱形，这极大地限制了归纳公式的片段。我们证明了\(\textsf{ALBA}\)对于模态 FM 框架的健全性，并表明\(\textsf{ALBA}\)在归纳公式上取得了成功，类似于现有的设置（Conradie 和 Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012；Zhao 2021，in: Cia-battoni, Pimentel, Queiroz（编辑）逻辑、语言、信息和计算，Springer International Publishing，Cham，2022）。