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We prove that if a finite (3 + 1)-free ordered set of height two has the fixed point property, then it is dismantlable by irreducibles. We provide an example of a finite (3 + 1)-free ordered set of height three with the fixed point property and no irreducible elements. We characterize the minimal automorphic ordered sets which are respectively (3 + 1)-free, (2 + 2)-free and N-free.
We find a syntactic characterization of the class $\mathrm{\mathbf{SUB}}(\mathcal{S})\cap\mathrm{Fin}$ of finite lattices embeddable into convexity lattices of a certain class of posets which we call star-like posets and which is a proper subclass in the class of N-free posets. The characterization implies that the class $\mathrm{\mathbf{SUB}}(\mathcal{S})\cap\mathrm{Fin}$ forms a pseudovariety.
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