A relational structure X is called reversible iff each bijective homomorphism from X onto X is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a consistent L ∞ω -theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. In particular, we characterize the reversible countable ultrahomogeneous graphs. 2010 MSC: 03C30, 03C52, 03C98, 05C63, 05C20,
A poset P is called reversible iff every bijective homomorphism f : P → P is an automorphism. Let W and W * denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the formwhere γ i ∈ Lim ∪{0} and n i ∈ ω, defining I α := {i ∈ I : α i = α}, for α ∈ Ord, and J γ := {j ∈ I : γ j = γ}, for γ ∈ Lim 0 , we prove that i∈I L i is a reversible poset iff α i : i ∈ I is a finite-to-one sequence, or there is γ = max{γ i : i ∈ I}, for α ≤ γ we have |I α | < ω, and n i : i ∈ J γ \ I γ is a reversible sequence of natural numbers. The same holds when L i ∈ W * , for all i ∈ I. In the general case, the reversibility of the whole union is equivalent to the reversibility of the union of components from W and the union of components from W * . 2010 MSC: 06A06, 06A05, 03E10, 03C07.
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