On Birkhoff's common abstraction problem

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a “common abstraction” that includes Boolean algebras and latticeordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA∨LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA∨LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA ∨ LG.