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International Conference on Modern Algebra in conjunction with the 17th annual Shanks Lectures
May 21-24, 2002
Vanderbilt University
Nashville, TN, USA

Organizers
Jonathan Farley, Ralph Freese, Matthew Gould, Peter Jipsen, George McNulty, Miklos Maroti, Alexander Ol'shanskii, Steven Tschantz, Constantine Tsinakis, Matthew Valeriote

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Relation algebras and multigroupoids
by
Mohamed El Bachraoui
Vrije Universiteit, Amsterdam, The Netherlands
Coauthors: Marcel van de Vel (Vrije Universiteit Amsterdam)

By a (semi-associative) multigroupoid we mean a relational structure the complex algebra of which is a (semi-associative) relation algebra. The complex algebra of a (semi-associative) multigroupoid U will be denoted by \mathfrakCm(U).

Given a finite (semi-associative) multigroupoid U = (U, C, r, I) and a (semi-associative) relation algebra \mathfrakR with universe R, the set U R of functions from U to R is the universe of a (semi-associative) relation algebra MU (\mathfrakR), called the U-(semi-associative) relation algebra over \mathfrakR. The composition of two functions f and g is defined at w in U as follows:
(f;g) (w) : = å
{f(u) ; g(v): u, v in U \textand  (u, v, w) in C }.
The composition is well-defined because U is finite. See . Note that this definition is a generalization of matrix relation algebras, where U is the standard multigroupoid on n2 pairs (i, j) ( i, j = 1, ... , n ), refer to. We use deg(\mathfrakR ) to denote the degree of \mathfrakR in the sense of Maddux, see . If U is a finite semi-associative multigroupoid and \mathfrakR is a semi-associative relation algebra then
deg(MU (\mathfrakR) ) = min { deg(\mathfrakCm(U) ), deg( \mathfrakR ) }.
Our definition of algebras of type MU (\mathfrakR) works if U is finite and \mathfrakR is arbitrary. It can also be adopted with minor changes if U is an arbitrary (semi-associative) multigroupoid and if \mathfrakR is a complete (semi-associative) relation algebra. We denote the obtained (semi-associative) relation algebra by U(\mathfrakR). The latter construction is clearly not a generalization of the former one. For this reason, different notations are used. One important result is: If U and V are semi-associative multigroupoids then
U( \mathfrakCm(V) ) \approx \mathfrakCm(U ×V ) \approx V( \mathfrakCm(U) ).
We will present more properties of these constructions.

Assume now that both U and \mathfrakR are arbitrary. Then both the completion \mathfrakR * and the perfect extension \mathfrakR+ of \mathfrakR are complete. We now can define two distinct (semi-associative) relation algebras:

In each situation, we could define an appropriate subalgebra generated by U R. (P1) Given a finite cardinal n, we showed in  that the class of n-matrix relation algebra is an elementary class. Is the class of U-relation algebras for some fixed multigroupoid U an elementary class?
(P2) Given a semi-associative multigroupoid U and a semi-associative relation algebra, what are the degrees of U * (\mathfrakR) and U+(\mathfrakR)?
(P3) Given two (semi-associative) relation algebras \mathfrakR and \mathfrakR', is there a realistic kind of product that generalizes all these constructions?

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El Bachraoui, M. and van de Vel, M., Matrix relation algebras, To appear in Alg. Univ.
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El Bachraoui, M. and van de Vel, M., Multigroupoids and relation algebras, In preparation.
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Maddux, R.D. A sequent calculus for relation algebras, Ann. Pure Appl. Logic, 25, 1983, pages: 73-101.

Date received: April 19, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-97.