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We deal with pairwise comparisons matrix with fuzzy elements (FPCM). Fuzzy elements are appropriate whenever the decision maker (DM) is uncertain about the value of his/her evaluation of the relative importance of elements in question, or, when aggregating crisp pairwise comparisons of a group of decision makers in the group DM problem. The problem is formulated in a general setting investigating pairwise comparisons matrices with elements from abelian linearly ordered group (alo-group). Such an approach enables extensions of traditional multiplicative, additive or fuzzy approaches. Continuing our research in \cite{Ramik2018}, here we propose a new order preservation concept based on alpha-cuts. Then we define an innovative concept of (weak) consistency of FPCMs, propose some desirable properties of priority vectors, and derive necessary and sufficient conditions for the existence of coherent vector (CV) and intensity vector (IV) of a FPCM. Finally, we formulate the optimization problem and derive the priority vector with the desirable properties. Illustrating examples are presented and discussed.