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In this paper we discuss different monotonicity definitions applied in stochastic modelling. Obviously, the relationships between the monotonicity concepts depends on the relation order that we consider on the state space. In the case of total ordering, the stochastic monotonicity used to build bounding models and the realizable monotonicity used in perfect simulation are equivalent to each other while in the case of partial order there is only implication between them. Indeed, there are cases of partial order, where we can't move from the stochastic monotonicity to the realizable monotonicity, this is why we will try to find the conditions for which there are equivalences between these two notions. In this study, we will present some examples to give better intuition and explanation of these concepts.