Abstract
We investigate geometric exchange properties in lattices of finite length that generalize the Steinitz exchange property of finite-dimensional vector spaces. In particular, we show that a stronger version of MacLane's exchange property for semimodular lattices is equivalent to the join-symmetric exchange property of Gaskill and Rival for modular lattices and, furthermore, that this exchange property characterizes strong semimodular lattices. An analogue of the basis exchange property for matroids is considered and seen to distinguish strong lattices in the class of semimodular lattices.
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