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We investigate an operator on classes of languages. For each class $C$, itoutputs a new class $FO^2(I_C)$ associated with a variant of two-variablefirst-order logic equipped with a signature$I_C$ built from $C$. For $C =\{\emptyset, A^*\}$, we get the variant $FO^2(<)$ equipped with the linearorder. For $C = \{\emptyset, \{\varepsilon\},A^+, A^*\}$, we get the variant$FO^2(<,+1)$, which also includes the successor. If $C$ consists of all Booleancombinations of languages $A^*aA^*$ where $a$ is a letter, we get the variant$FO^2(<,Bet)$, which also includes "between relations". We prove a genericalgebraic characterization of the classes $FO^2(I_C)$. It smoothly andelegantly generalizes the known ones for all aforementioned cases. Moreover, itimplies 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 unarytemporal logic. For each class $C$, we consider a variant $TL(C)$ of unarytemporal logic whose future/past modalities depend on $C$ and such that $TL(C)= FO^2(I_C)$. Finally, we also characterize $FL(C)$ and $PL(C)$, thepure-future and pure-past restrictions of $TL(C)$. These characterizations aswell imply that if \Cs is a class with decidable separation, then $FL(C)$ and$PL(C)$ have decidable membership.

A generic characterization of generalized unary temporal logic and two-variable first-order logic