We present some results concerning elimination of quantifiers and elementary equivalence for Boolean products of real closed valuation rings and fields. We also study rings of continuous functions and rings of definable functions over real closed valuation rings under this point of view.

The paper shows elimination of imaginaries for real closed valued fields to suitable sorts. We also show that this result is in some sense optimal. The paper includes a quantifier elimination theorem for real closed valued fields in a language with sorts for the field, value group and residue field.

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

We explore analogues of o-minimality and weak o-minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ∅ there are differences. Our main result is a structure theory for ℵ0-categorical weakly circularly minimal structures. There is a 5-homogeneous example which is not 6-homogeneous, but any example which is k-homogeneous for some k ≥ 6 is k-homogeneous for all k.

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

Let $$ {\mathcal {M}}=(M, <, \ldots ) $$ be a weakly o-minimal structure. Assume that $$ {\mathcal {D}}ef({\mathcal {M}})$$ is the collection of all definable sets of $$ {\mathcal {M}} $$ and for any $$ m\in {\mathbb {N}} $$, $$ {\mathcal {D}}ef_m({\mathcal {M}}) $$ is the collection of all definable subsets of $$ M^m $$ in $$ {\mathcal {M}} $$. We show that the structure $$ {\mathcal {M}} $$ has the strong cell decomposition property if and only if there is (...) an o-minimal structure $$ {\mathcal {N}} $$ such that $$ {\mathcal {D}}ef({\mathcal {M}})=\{Y\cap M^m: \ m\in {\mathbb {N}}, Y\in {\mathcal {D}}ef_m({\mathcal {N}})\} $$. Using this result, we prove that: (a) Every induced structure has the strong cell decomposition property. (b) The structure $$ {\mathcal {M}} $$ has the strong cell decomposition property if and only if the weakly o-minimal structure $$ {\mathcal {M}}^*_M $$ has the strong cell decomposition property. Also we examine some properties of non-valuational weakly o-minimal structures in the context of weakly o-minimal structures admitting the strong cell decomposition property. (shrink)

Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an o-minimal trace.

In o-minimal structures, every cell is definably connected and every definable set is a finite union of its definably connected components. In this note, we introduce pseudo definably connected definable sets in weakly o-minimal structures having strong cell decomposition, and prove that every strong cell in those structures is pseudo definably connected. It follows that every definable set can be written as a finite union of its pseudo definably connected components. We also show that the projections of pseudo definably connected (...) definable sets are pseudo definably connected. Finally, we compare pseudo definable connectedness with weak definable connectedness of definable sets in weakly o-minimal structures. (shrink)

Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the full first order theory of over any set. We also show that if two weakly o‐minimal structures with the strong cell decomposition property are isomorphic then, their canonical o‐minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o‐minimal theory with prime models.

Let M=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}=}$$\end{document} be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} satisfy an extended (...) version of IVP. After introducing a weak version of definable connectedness in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document}, we prove that strong cells in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} are weakly definably connected, so every set definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} is a finite union of its weakly definably connected components, provided that M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph which was examined for definable functions in and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} having cell decomposition, satisfies NDG, i.e. every definable function in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has no dense graph. (shrink)