The many descendants of Tarski’s Relation Algebras

Peter Jipsen

Vanderbilt University

Alfred Tarski Centenary Conference, Warsaw, May 29, 2001

A story about the creation of Relation Algebras

In the beginning there were algebras of concrete relations.

Tarski saw they were good, and he separated the interesting ideas from the trivial ones.

And Tarski said “Let there be an abstract theory about these algebras”.

So he made the theory of Relation Algebras. And he saw it was good.

And then Tarski said “Let the theory produce all the known results about concrete relations”. And it was so.

And he proved many interesting new results about relation algebras, including a correspondence with 3-variable logic that allowed the interpretation of set theory and he provided the first example of an undecidable equational theory.

And Tarski said “Let the minds teem with new conjectures, let ideas fly, and let the community produce many new related theories and results”.

Thus the field of relation algebras was born, with its many applications and connections to other areas.

(all quotes fictitious; passage based on well known source)

And he did not rest. The trinity of Henkin, Monk, Tarski wrote two volumes about Cylindric Algebras (including several chapters on general algebra and relation algebras).

His many disciples worked tirelessly to spread the word.

And there was a cultural upheaval that made Tarski’s name spread far and wide: computer science emerged as a major discipline.

Some statistics about Tarski:

Number of authored papers in MR:

Tarski 125

Erdös 1535 (the most publications by any author)

Number of reviews mentioning name in MR:

Tarski 2133

Erdös 6878

Number of web pages mentioning name (on Google):

Tarski 36000

Erdös 30000

Number of papers in 11 major Mathematics journals that mention Tarski: 1041

Number of papers in 10 major Philosophy journals that mention Tarski: 1047

List of the journals searched in JSTOR:

Mathematics (11 journals)

1. American Journal of Mathematics (1878-1995)

2. American Mathematical Monthly (1894-1995)

3. Annals of Mathematics (1884-1995)

4. Journal of Symbolic Logic (1936-1996)

5. Journal of the American Mathematical Society (1988-1995)

6. Mathematics of Computation (1960-1995)

7. Proceedings of the American Mathematical Society (1950-1995)

8. SIAM Journal on Applied Mathematics (1966-1995)

9. SIAM Journal on Numerical Analysis (1966-1995)

10. SIAM Review (1959-1995)

11. Transactions of the American Mathematical Society (1900-1995)

Philosophy (10 journals)

1. Ethics (1938-1995)

2. Journal of Philosophy (1921-1995)

3. Journal of Symbolic Logic (1936-1996)

4. Mind (1876-1993)

5. Nous (1967-1995)

6. Philosophical Perspectives (1987-1995)

7. Philosophical Quarterly (1950-1995)

8. Philosophical Review (1892-1997)

9. Philosophy and Phenomenological Research (1940-1995)

10. Philosophy and Public Affairs (1971-1995)

Coauthors of Alfred Tarski

(From: Erdos1, Version 2001, January 30, 2001

This is a list of the 507 co-authors of Paul Erdos, together with their co-authors listed beneath them. The date of first joint paper with Erdos is given, followed by the number of joint publications (if it is more than one). An asterisk following the name indicates that this Erdos co-author is known to be deceased. Please send corrections and comments to <grossman@oakland.edu>.)

TARSKI, ALFRED* 1943: 2

Andreka, Hajnal

Banach, Stefan

Beth, Evert W.

Chang, Chen Chung

Chin, Louise H.

Doner, John E.

Erdos, Paul

Fell, James M. G.

Givant, Steven R.

Henkin, Leon A.

Horn, Alfred

Jonsson, Bjarni

Keisler, H. Jerome

Kuratowski, Kazimierz

Lindenbaum, A.

Maddux, Roger D.

McKinsey, J. C. C.

Monk, J. Donald

Mostowski, Andrzej

Nemeti, Istvan

Schwabhauser, Wolfram

Scott, Dana S.

Sierpinski, Waclaw

Smith, Edgar C., Jr.

Szczerba, Leslaw W.

Szmielew, Wanda

Vaught, Robert L.

A real quote (according to MacTutor) of Tarski:

“You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflict”

A tour of theories and structures close to relation algebras. We start with the variety RA.

Here is the definition of Relation Algberas as recorded by Bjarni Jonsson in a seminar that Alfred Tarski gave at Berkeley in the 1940s. Note that the equational axiomatisation is not chosen as the original definition, but rather it is derived from the more useful and compact quasi-equational definition.

First some preliminaries: (A groupoid is what is nowadays called a monoid)