## On variants of $$o$$-minimality.(English)Zbl 0858.03039

A $$C$$-structure $$(M,C)$$ consists of a structure $$M$$ along with a ternary relation $$C(x;y,z)$$ on the domain of $$M$$, where the domain of $$M$$ has at least 2 elements and $$C$$ satisfies certain additional axioms. Let $$(T,\leq)$$ be an infinite tree in which there are incomparable elements above any node of the tree. The set $$M$$ of maximal chains of $$T$$ with the ternary relation $$C(x;y,z)$$ induced on $$M$$ by requiring that $$y$$ and $$z$$ branch above where $$x$$ and $$y$$ branch turns out to be a very representative $$C$$-structure. A $$C$$-minimal structure is an expansion $${\mathcal M} = (M,C, \dots)$$ of a $$C$$-structure $$(M,C)$$ such that for every $${\mathcal N} \equiv {\mathcal M}$$, every definable subset of $$N$$ is quantifier-free definable in $$(N,C)$$.
This paper is an investigation of $$C$$-minimal structures as an interesting variant of $$o$$-minimality. This choice is influenced by work of Adeleke and Neumann on Jordan groups. The authors develop some general model theory for $$C$$-minimal structures. They go on to study $$C$$-minimal groups and fields. The paper ends with some other variants of $$o$$-minimality.

### MSC:

 03C50 Models with special properties (saturated, rigid, etc.) 03C60 Model-theoretic algebra
Full Text:

### References:

 [1] S. Adeleke and D. Macpherson, Classification of infinite primitive Jordan permutation groups, Proc. London Math. Soc., to appear. · Zbl 0839.20002 [2] S. Adeleke and P.M. Neumann, Primitive permutation groups with primitive Jordan sets, J. London Math. Soc., to appear. · Zbl 0865.20005 [3] S. Adeleke and P.M. Neumann, Relations related to betweenness: their structure and automorphisms, Memoirs Amer. Math. Soc., to appear. · Zbl 0896.08001 [4] Aschbacher, M., Finite group theory, (1986), Cambridge Univ. Press Cambridge · Zbl 0583.20001 [5] Cameron, P.J., Orbits of permutation groups on unordered sets, IV: homogeneity and transitivity, J. London math. soc., 27, 2, 238-247, (1983) · Zbl 0519.20004 [6] Cameron, P.J., Some treelike objects, Q. J. math. Oxford, 38, 2, 155-183, (1987) · Zbl 0628.05039 [7] Cameron, P.J., Oligomorphic permutation groups, () · Zbl 1205.20002 [8] Droste, M., Structure of partially ordered sets with transitive automorphism groups, () · Zbl 0574.06001 [9] Droste, M.; Holland, W.C.; Macpherson, D., Automorphism groups of infinite semilinear orders (I), (), 454-478 · Zbl 0636.20003 [10] Fraissé, R.; Fraissé, R., Sur certains relations qui généralisent l’ordre des nombres rationnels, C. R. acad. sci. Paris, C. R. acad. sci. Paris, 237, 542, (1953) · Zbl 0051.03803 [11] Fraissé, R., Theory of relations, (1986), North-Holland Amsterdam · Zbl 0593.04001 [12] Fuchs, L., Partially ordered algebraic systems, (1963), Pergamon Press Oxford · Zbl 0137.02001 [13] Hall, P.; Kulatilaka, C.R., A property of locally finite groups, J. London math. soc., 39, 235-239, (1964) · Zbl 0136.27903 [14] Haskell, D.; Macpherson, D., Cell decompositions of C-minimal structures, Ann. pure appl. logic, 66, 113-162, (1994) · Zbl 0790.03039 [15] Hodges, W.A.; Hodkinson, I.M.; Macpherson, H.D., Relative categoricity, omega-categoricity, and co-ordinatisation, Ann. pure appl. logic, 46, 169-200, (1990) · Zbl 0699.03016 [16] Kantor, W.M.; Liebeck, M.W.; Macpherson, H.D., $$ℵ0- categorical$$ structures smoothly approximated by finite substructures, (), 439-463, (3) · Zbl 0649.03018 [17] Knight, J.F.; Pillay, A.; Steinhorn, C., Definable sets in ordered structures II, Trans. amer. math. soc., 295, 593-605, (1986) · Zbl 0662.03024 [18] D.W. Kueker and P. Steitz, Definability over sets, preprint. [19] D.W. Kueker and P. Steitz, Stabilizers of definable sets in saturated models, preprint. [20] Macpherson, D., A survey of Jordan groups, (), 73-110 · Zbl 0824.20003 [21] Pillay, A., An introduction to stability theory, (1983), Oxford Univ. Press Oxford · Zbl 0526.03014 [22] Pillay, A., First order topological structures and theories, J. symbolic logic, 52, 763-778, (1987) · Zbl 0628.03022 [23] Pillay, A.; Steinhorn, C., Definable sets in ordered structures I, Trans. amer. math. soc., 295, 565-592, (1986) · Zbl 0662.03023 [24] Razenj, V., One dimensional groups over an o-minimal structure, Ann. pure appl. logic, 53, 269-277, (1991) · Zbl 0747.03016 [25] Rieger, L.S., On the ordered and cyclically ordered groups I, Vĕstník král. Cĕsḱ$$spol. nauk, 1-31, (1946$$ [26] Rieger, L.S., On the ordered and cyclically ordered groups II, Vĕstník král. Cs̆ké spol. nauk, 1-33, (1947) [27] Rieger, L.S., On the ordered and cyclically ordered groups III, Vĕstník král. Cĕské spol. nauk, 1-26, (1948), (in Czech) [28] Rubin, M., On the reconstruction of $$ℵ0- categorical$$ structures from their automorphism group, (), 225-249, (3) · Zbl 0799.03037 [29] Sacks, G., Saturated model theory, (1972), Benjamin Reading, MA · Zbl 0242.02054 [30] Shelah, S., Classification theory and the number of non-isomorphic models, (1990), North-Holland Amsterdam · Zbl 0713.03013 [31] Swierczkowski, S., On cyclically ordered groups, Fundam. math., 47, 161-166, (1959) · Zbl 0096.01501 [32] Thomas, S.R., Groups acting on infinite-dimensional projective spaces, J. London math. soc., 34, 2, 265-273, (1986) · Zbl 0606.20005 [33] Weispfenning, V., Quantifier elimination and decision procedures for valued fields, (), 419-472 [34] Zelmanov, E.I., The solution of the restricted Burnside problem for groups of odd exponent, Izv. akad. nauk USSR, 54, 42-59, (1990) · Zbl 0704.20030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.