On Birkhoff’s Common Abstraction Problem

Studia Logica 100 (6):1079-1105 (2012)


In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a "common abstraction" that includes Boolean algebras and lattice-ordered 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 and their join ∨ in the lattice of subvarieties of ℒ (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for ∨ relative to ℒ. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing ∨

Download options


    Upload a copy of this work     Papers currently archived: 72,891

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library


Added to PP

35 (#330,102)

6 months
1 (#386,040)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Francesco Paoli
Universita di Cagliari

Citations of this work

No citations found.

Add more citations

Similar books and articles

The Birth of Quantum Logic.Miklós Rédei - 2007 - History and Philosophy of Logic 28 (2):107-122.
Bad Company Tamed.Øystein Linnebo - 2009 - Synthese 170 (3):371 - 391.
A Strengthening of the Caesar Problem.Joongol Kim - 2011 - Erkenntnis 75 (1):123-136.
Strategies of Abstraction.Richard Levins - 2006 - Biology and Philosophy 21 (5):741-755.
Birkhoff Completeness in Institutions.Mihai Codescu & Daniel Găină - 2008 - Logica Universalis 2 (2):277-309.
The Limits of Abstraction.Kit Fine - 2002 - Oxford, England: Oxford University Press.