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Monolayer and two-dimensional (2D) systems exhibit rich phase behavior, compared with 3D systems, in particular, due to the hexatic phase playing a central role in melting scenarios. The attraction range is known to affect critical gas-liquid behavior (liquid-liquid in protein and colloidal systems), but the effect of attraction on melting in 2D systems remains unstudied systematically. Here, we reveal how the attraction range affects the phase diagrams and melting scenarios in a 2D system. Using molecular dynamics simulations we considered the generalized Lennard-Jones system with a fixed repulsion branch and different power indices of attraction, from long-range dipolar to short-range sticky-spheres-like. A drop in the attraction range has been found to reduce the temperature of the gas-liquid critical point, bringing it closer to the gas-liquid-solid triple point. At high-temperatures, attraction does not affect the melting scenario that proceeds through the cascade of solid-hexatic (Berezinski-Kosterlitz-Thouless) and hexatic-liquid (first-order) phase transitions. In the case of dipolar attraction, we observed \emph{two triple points}, inherent in a 2D system: hexatic-liquid-gas and crystal-hexatic-gas, the temperature of the crystal-hexatic-gas triple point is \emph{below} the hexatic-liquid-gas triple point. This observation may have far-reaching consequences for future studies, since phase diagrams determine possible routes of self-assembly in molecular, protein, and colloidal systems, whereas the attraction range can be adjusted with complex solvents and external electric or magnetic fields. The results obtained may be widely used in condensed matter, chemical physics, materials science, and soft matter., Comment: 11 pages, 11 figures