The author gives a precise specification of the Birkhoff-Frink representation of lattices (see G. Birkhoff and O. Frink [Trans. Am. Math. Soc. 64, 299-316 (1948; Zbl 0032.00504)]). By this representation every lattice is isomorphic to a lattice whose elements are sets of sets whose meet operation is defined as intersection and whose join operation \(\vee^*\) is defined as \[ A\vee^*B=A\cup B\cup\{Z:(\exists X\in A)(\exists Y\in B)X\cap Y\subset Z\}. \] This is the specification of the Birkhoff-Frink representation given by the author, and it corresponds to the semantic clause for disjunction introduced previously by the author [Stud. Logica 48, 41-65 (1989; Zbl 0688.03012)]. It also enables one to connect this representation with Stone’s representation theory for distributive lattices.
For the entire collection see [Zbl 0977.00051].