arXiv subjects
Publications and source records attributed to Gérard Leloup.
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove several properties.
We make available some results about model theory cyclically ordered groups. We start with a classification of complete theories of divisible abelian cyclically ordered groups. Then we look at the cyclically ordered groups where the only parametrically definable subsets are finite unions of singletons and open intervals, and those where the definable subsets are finite union of singletons and c-convex subsets, where being c-convex is the analogue of being convex in the linearly ordered case.
We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.
We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex numbers, that every discrete c-regular cyclically ordered abelian group is elementarily equivalent to some ultraproduct of finite cyclic groups, and that the discrete regular non-c-regular cyclically ordered abelian groups are elementarily equivalent to the linearly cyclically ordered group of integers.
By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group $H$ a pair $(G,z)$ where $G$ is a totally ordered group and $z$ is an element in the center of $G$, generating a cofinal subgroup $\langle z\rangle$ of $G$, and such that the quotient group $G/\langle z\rangle$ is isomorphic to $H$. We first establish that, in this correspondence, the first order theory of the cyclically ordered group $H$ is uniquely determined by the first order theory of the pair $(G,z)$. Then we prove that the class of cyclically ordered groups is an elementary class and give an axiom system for it. Finally we show that, in opposition to the fact that all theories of totally Abelian ordered groups have the same universal part, there are uncountably many universal theories of Abelian cyclically ordered groups. We give for each of these universal theories an invariant, which is a pair of subgroups of the group of unimodular complex numbers.