Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

We present a new method of constructing Condorcet domains from pairs of Condorcet domains of smaller sizes (concatenation + shuffle scheme). The concatenation + shuffle scheme provides maximal, connected, copious, peak-pit domains whenever the original domains have these properties. It allows to construct maximal peak-pit Condorcet domains that are larger than those obtained by the Fishburn’s alternating scheme for all $$n\ge 13$$ n ≥ 13 alternatives. For a large number n of alternatives, we get a lower bound $$2.1045^{n}$$ 2. 1045 (...) n for the cardinality of the largest peak-pit Condorcet domain and a lower bound $$2.1890^{n}$$ 2. 1890 n for the cardinality of the largest Condorcet domain, improving Fishburn’s result. We also show that all Arrow’s single-peaked domains can be constructed by concatenation + shuffle scheme starting from the trivial domain. (shrink)

We show that, when the number of participating agents n tends to infinity, all classical social choice rules are asymptotically strategy -proof with the proportion of manipulable profiles being of order O.