Left modularity and extremality are equivalent for well-separated kappa-lattices and for weakly atomic completely semidistributive lattices, and the torsion-class lattice of an algebra is left modular exactly when the algebra is brick-directed.
Thomas, An analogue of distributivity for ungraded lattices, Order 23 (2006), no
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Left modularity and extremality for (some) infinite lattices
Left modularity and extremality are equivalent for well-separated kappa-lattices and for weakly atomic completely semidistributive lattices, and the torsion-class lattice of an algebra is left modular exactly when the algebra is brick-directed.