A simple algebraic characterization of weak equational theories
by
Grzegorz Binczak
Institute of Mathematics, Warsaw University

In my talk I show some new algebraic characterization of weak equational theories. It is simpler than characterizations in [1] and [2]. A family \Theta = (\theta_i)_{i in I} is an initial family in a partial algebra A iff for every i in I, \theta_i is a closed congruence on a relative subalgebra dom\theta_i of A, where dom\theta_i is a nonempty initial segment in A. Then we can define a convolution Cnv(\Theta)={(a, b) in A²\colon for all_{i in I}a, b in dom\theta_i ===> (a, b) in \theta_i}. I prove that for every nonempty set of variables A a subset T of a set T_F(A)² of all pairs of terms (over A) is a weak equational theory (over A) of some class of partial algebras iff T is a fully invariant convolution of some initial family in T_F(A).

References

[1] H. Höft "Weak and strong equations in partial algebras", Algebra Universalis, 3 (1973) 203-215

[2] L. Rudak Ä completness theorem for weak equational logic" Algebra Universalis, 16 (1983) 331-337

Date received: May 18, 2000

Copyright © 2000 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 # caee-42.