The E-base is valid and minimum for finite semidistributive closure lattices, exact characterizations are given for modular and geometric lattices, and every finite lattice embeds into a lattice with valid E-base.
Independence of Essential Sets in Finite Implication Bases
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abstract
A new characterization is given to describe implication bases of a closure system in terms of the system's quasi-closed sets. Using this characterization, it is possible to show that groups of implications corresponding to distinct essential sets are interchangeable across different bases. It follows from this result that the sum of cardinalities of right sides of all implications corresponding to a single essential set in an optimal basis is fixed, solving an open conjecture by K. Adaricheva and J.B. Nation in 2014. These results provider greater insight into the global structure of implication bases.
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On the $E$-base of Finite Lattices: Semidistributive, Modular, and Geometric Lattices
The E-base is valid and minimum for finite semidistributive closure lattices, exact characterizations are given for modular and geometric lattices, and every finite lattice embeds into a lattice with valid E-base.