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Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership

We study two extensions of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, "the letter $a$ appears between positions $x$ and $y$" and "the factor $u$ appears between&nbsp;[&hellip;]