nLab quasiminimal structure

Idea

A quasiminimal structure is a particular class of pregeometries (infinite finitary matroids). Quasiminimal excellent classes were defined by Zilber as useful axiomatic setup satisfying categoricity.

Literature

Introduced in

• Boris Zilber, A categoricity theorem for quasi-minimal excellent classes, Logic and its applications, Contemporary Mathematics 380 (Amer. Math. Soc. 2005) 297–306.

In original treatments, among the axioms, excellence was hard/technical to prove. A breakthrough is a result that it is essentially automatic:

• Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesälä, Jonathan Kirby, Quasiminimal structures and excellence, Bull. London Math. Soc. 46 (2014) 155–163 doi
• Levon Haykazyan, Categoricity in quasiminimal pregeometry classes, J. Symb. Logic 81(1), 56–64 (2014) doi arXiv:1308.1892

Applications include

• B. Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero, J. London Math. Soc. (2) 74 (2006) 41–58
• B. Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic 132 (2005) 67–95

Other

• John T. Baldwin, Notes on quasiminimality and excellence, The Bulletin of Symbolic Logic 10:3 (2004) 334–366

category: model theory