Our Monte Carlo ͑MC͒ method has been extended to determine the type of ordering, the character of phase transitions, and positions of the tricritical points in the three-dimensional Ashkin-Teller spin-lattice model on a cubic lattice, which is an example of a system with multicomponent order parameter. The precise and detailed analyses of the system's behavior at the ends of the lines of Ising-like phase transitions are performed and the corrected phase diagram in these regions is presented and discussed. The localizations of the tricritical points are also given. The specific behavior of the Binder cumulant when crossing a bifurcation point has been discovered and interpreted. Some general conclusions on methods of investigation of the critical behavior of a system with a multicomponent order parameter are closing the paper.

The Monte Carlo simulations in the 3D Ashkin -Teller model on a cubic lattice are performed. The study is undertaken in the region where the universality class of the phase transitions has not been unambigously resolved yet. Using the finite-size scaling relation between the magnetization, the temperature and the size of the system, the method of calculation of the critical exponent y h is proposed. Our preliminary results obtained for y h suggest a nonuniversal behavior, similarly as it was observed in the 2D case. Its value seems to change continuously in some interval approaching the Ising value near to all the tricritical points except one.

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