A relational structure is called reversible iff each bijective endomorphism (condensation) of that structure is an automorphism. We show that reversibility is an invariant of some forms of L∞ω −bi-interpretability, implying that the condensation monoids of structures are topologically isomorphic. Applying these results, we prove that, in particular, all orbits of ultrahomogeneous tournaments and reversible ultrahomogeneous m-uniform hypergraphs are reversible relations and that the same holds for the orbits of reversible ultrahomogeneous digraphs definable by formulas which are not R-negative.

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

We extend Ahlbrandt and Ziegler’s reconstruction results ([1]) to the metric setting: we show that separably categorical structures are determined, up to bi-interpretability, by their automorphism groups.

Many classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

We show that R.J. Thompson’s groups F and T are bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the group T provides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].

We investigate definability in $\mathcal{R}$, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove that the recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes $\mathrm{Low}_n$ and $\mathrm{High}_{n-1}$$(n\ge 2)$ are definable in $\mathcal{R}$ without parameters and, more generally, that all relations on $\mathcal{R}$ that are definable in arithmetic and invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare's Question 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), Chapter XVI).

1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.