Symmetric revealed cores and pseudocores, and Lawvere-Tierney closure operators

A choice function is a symmetric revealed core if there exists a symmetric irreflexive ‘dominance’ digraph such that choice sets consist precisely of the locally undominated outcomes of the latter. Symmetric revealed pseudocores are similarly defined by omitting the irreflexivity requirement on the underlying digraph. Lawvere-Tierney (LT) closure operators are those closure operators which are meet-homomorphic: they may be regarded as an algebraic representation of a geometric modality denoting ‘locally true’, and provide the mathematical backbone of a generalized version of so-called ‘Grothendieck topologies’ in categories. The classes of symmetric revealed cores and pseudocores are characterized, and their basic order-theoretic structure is studied. In particular, it is shown that their respective posets are sub-meet-semilattices of the canonical lattice of choice functions. An order duality theorem concerning the posets of symmetric revealed pseudocores and LT closure operators on a given ground set is also established.