A New w-Stable Plane

Gianluca Paolini

Abstrakt

We use a variation on Mason’s α-function as a pre-dimension function to construct a not one-based ω–stable plane P (i.e. a simple rank 3 matroid) which does not admit an algebraic representation (in the sense of matroid theory) over any field. Furthermore, we characterize forking in Th(P), we prove that algebraic closure and intrinsic closure coincide in Th(P), and we show that Th(P) fails weak elimination of imaginaries, and has Morley rank ω.

AMS Subject Classification: 03C45, 05B35

Słowa kluczowe: Hrushovski constructions, matroids, w-stable structures
References

[1] M. Aigner, Combinatorial Theory, Springer-Verlag, Berlin Heidelberg, 1979.

[2] J. Baldwin, G. Paolini, Strongly Minimal Steiner Systems I: Existence, To appear in J. Symb. Logic, available at: https://arxiv.org/abs/1903.03541.

[3] J. Baldwin, Strongly Minimal Steiner Systems II: Coordinatization and Strongly Minimal Quasigroups, In preparation.

[4] J. Baldwin, Niandong Shi, Stable Generic Structures, Ann. Pure Appl. Logic 79:1 (1996), 1–35.

[5] H.H. Crapo, and G. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, M.I.T. Press, Cambridge, Mass, 1970.

[6] H. H. Crapo, Single-Element Extensions of Matroids, J. Res. Nat. Bur. Standards Sect. B, v. 69B (1965), 55–65. MR 32 # 7461.

[7] D. Evans, Matroid Theory and Hrushovski’s Predimension Construction, available at: https://arxiv.org/abs/1105.3822.

[8] D. Evans, An Introduction to Ampleness, available at: http://wwwf.imperial.ac.uk/~dmevans/OxfordPGMT.pdf.

[9] A. Hasson, O. Mermelstein, Reducts of Hrushovski’s Constructions of a Higher Geometrical Arity, Fund. Math., to appear.

[10] Ehud Hrushovski, A New Strongly Minimal Set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147-166.

[11] T. Hyttinen, G. Paolini, Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices, Ann. Pure Appl. Logic 169:2 (2018), 117–145.

[12] J.P.S. Kung, A Source Book in Matroid Theory, Birkh¨auser Boston, Inc., Boston, MA, 1986.

[13] B. Lindstr¨om, A Class of non-Algebraic Matroids of Rank Three, Geom. Dedicata 23:3 (1987), 255–258.

[14] B. Lindstr¨om, A Desarguesian Theorem for Algebraic Combinatorial Geometries, Combinatorica 5:3 (1985), 237–239.

[15] D. Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, 217, Springer-Verlag, New York, 2002.

[16] J.H. Mason, On a Class of Matroids Arising from Paths in Graphs, Proc. London Math. Soc. (3) 25 (1972), 55–74.

[17] O. Mermelstein, An Ab Initio Construction of a Geometry, available at: https: //arxiv.org/abs/1709.07353.

[18] G. Paolini, A Universal Homogeneous Simple Matroid of Rank 3, Bol. Mat. (UNAL, Colombia) 25:1 (2018), 39–48.

[19] K. Tent, M. Ziegler, A Course in Model Theory, Lecture Notes in Logic, Cambridge University Press, 2012.

[20] D.J.A. Welsh, Matroid Theory, L. M. S. Monographs, No. 8. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976.

[21] N. White (ed.), Combinatorial Geometries, Encyclopedia of Mathematics and its Applications, 29. Cambridge University Press, Cambridge, 1987.

[22] M. Ziegler, An Exposition of Hrushovski’s New Strongly Minimal Set, Ann. Pure Appl. Logic 164:12 (2013), 1507–1519.

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