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In this paper, we consider the Potts-SOS model where the spin takes values inthe set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all thetranslation-invariant splitting Gibbs measures for this model in someconditions. Moreover, we investigate whether these Gibbs measures are extremalor non-extremal in the set of all Gibbs measures.

Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree