ADDITIONAL NOTES RE

George M. Bergman, Constructing division rings as module-theoretic direct limits Trans. A. M. S. 354 (2002) 2079-2114.

  • It turns out that the concept which (as mentioned in this paper) is called a "dependence structure" in developments of the parallel properties of linear dependence and algebraic dependence, and a "matroid structure" in the combinatorial literature, has a third name: Model theorists call it a "pregeometry".  One can find many examples by doing a MathSciNet search for that word in Subject Classification 03. (One must restrict to that Subject Classification - or at least exclude Subject Classifications 81 and 83 - to avoid the more numerous occurrences of "pregeometry", in what I assume is a different sense, in quantum mechanics.)

  • Additional reference: Some work related to the idea sketched in the last section, of constructing matroidal structures on objects of varieties of algebras (in the sense of universal algebra) other than modules, can be found in

    John Fountain and Victoria Gould, Relatively free algebras with weak exchange properties, J. Aust. Math. Soc. 75 (2003) 355--384.

    and papers referred to there.  The closure operators they consider are  <X> =  subalgebra generated by X, and  PC(X)  ("pure closure of  X")  = { y | <{y}>  ∩ <X>  is nontrivial },  in the latter case restricting attention to algebras on which PC is a closure operator. 

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