We investigate an operator on classes of languages. For each class $C$, it outputs a new class $FO^2(I_C)$ associated with a variant of two-variable first-order logic equipped with a signature$I_C$ built from $C$. For $C = \{\emptyset, A^*\}$, we get the variant $FO^2(<)$ equipped with the linear order. For $C = \{\emptyset, \{\varepsilon\},A^+, A^*\}$, we get the variant $FO^2(<,+1)$, which also includes the successor. If $C$ consists of all Boolean combinations of languages $A^*aA^*$ where $a$ is a letter, we get the variant $FO^2(<,Bet)$, which also includes "between relations". We prove a generic algebraic characterization of the classes $FO^2(I_C)$. It smoothly and elegantly generalizes the known ones for all aforementioned cases. Moreover, it implies that if $C$ has decidable separation (plus mild properties), then $FO^2(I_C)$ has a decidable membership problem. We actually work with an equivalent definition of \fodc in terms of unary temporal logic. For each class $C$, we consider a variant $TL(C)$ of unary temporal logic whose future/past modalities depend on $C$ and such that $TL(C) = FO^2(I_C)$. Finally, we also characterize $FL(C)$ and $PL(C)$, the pure-future and pure-past restrictions of $TL(C)$. These characterizations as well imply that if \Cs is a class with decidable separation, then $FL(C)$ and $PL(C)$ have decidable membership.

翻译：我们研究语言类上的一个算子。对于每个类$C$，该算子输出一个新类$FO^2(I_C)$，其关联于一个带由$C$构建的签名$I_C$的二变量一阶逻辑变体。当$C = \{\emptyset, A^*\}$时，我们得到带线性序的变体$FO^2(<)$。当$C = \{\emptyset, \{\varepsilon\},A^+, A^*\}$时，我们得到包含后继关系的变体$FO^2(<,+1)$。若$C$由所有形如$A^*aA^*$（其中$a$为字母）的语言的布尔组合构成，则得到包含"介于关系"的变体$FO^2(<,Bet)$。我们证明类$FO^2(I_C)$具有通用的代数刻画，该刻画平滑优雅地推广了上述所有已知情形。进一步，该刻画蕴含：若$C$具有可判定的分离性质（加上温和条件），则$FO^2(I_C)$具有可判定的成员资格问题。我们实际上利用等价的一元时态逻辑定义来开展工作。对于每个类$C$，我们考虑一元时态逻辑的变体$TL(C)$，其未来/过去模态依赖于$C$，且满足$TL(C) = FO^2(I_C)$。最后，我们还刻画了$FL(C)$和$PL(C)$——即$TL(C)$的纯未来和纯过去限制。这些刻画同样表明：若$C$是具有可判定分离性质的类，则$FL(C)$和$PL(C)$具有可判定的成员资格。