Relational lattices via duality

Relational lattices were introduced by Spight and Tropashko in a series of preprints [6,7,8]; equational principles were shown to hold in a signature strictly larger than the one of lattice theory. Litak, Mikulás and Hidders [2] exhibited later two non-trivial lattice identities holding in this class. Relational lattices are interesting in view of their close connection with database theory. An element of the relational lattice R(D,A) is a table whose entries are from D and whose header is a subset of A, a total set of attributes; mathematically, a table is a pair (X,T) with X ⊆ A and T ⊆ DX. Given two tables (Xi, Ti), i ∈ 2, the order is given by (X0, T0) ≤ (X1, T1) iff X1 ⊆ X0 and T0 ⊆ {f | fX1 ∈ T1}. This ordered set is a lattice whose operations coincide with two well-known operations in the theory and practice of databases. The meet is the outer-join of tables: (X0, T0) ∧ (X1, T1) = (X0 ∪X1, T), T = {f ∈ DX0∪X1 | fXi ∈ Ti, i = 0, 1}, while the join is the inner-union of tables: (X0, T0) ∨ (X1, T1) = (X0 ∩X1, T)