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Extending Algebras to Model Congruence Schemes

Published online by Cambridge University Press:  20 November 2018

J. Berman ,
G. Grätzer  and
C. R. Platt
J. Berman
Affiliation:
University of Illinois at Chicago, Chicago, Illinois
G. Grätzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba
C. R. Platt
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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This paper is concerned with the description of principal congruence relations. Given elements a and b of a universal algebra , let θ(a, b) denote the smallest congruence relation on containing the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says that c ≡ d(θ(a, b)) if and only if there exists a sequence z0, z1, …, zn of elements of and a sequence f1, f2, …, fn of unary algebraic functions such that c = z0, d = zn, and for each i = 1, …, n,

Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice of a, b, c and d. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 2 , 01 April 1986 , pp. 257 - 276
Copyright
Copyright © Canadian Mathematical Society 1986

References

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