Abstract
This paper gives characterisations of properties of varieties such as congruence-distributivity (CD), filtrality (FI), CD and having complemented principal congruences — these last two properties are shown to be equivalent- and having restricted equationally definable principal congruences (REDPC), in terms of the existence of some kind of polynomials. These are generalisations both of Jónsson's famous theorem characterizing CD as well as results concerning the (dual) discriminator. The methods are applied to show that REDPC implies CD, which was a problem asked in [2]. A generalisation of the concept of the Mal'cev condition — the so called Pixley-condition — is defined, and it is shown that filtrality and REDPC are Pixley-conditions. The relations between several concepts connected with the above ones are also investigated.
The definitions can be found in section 2, and the results are contained in section 4, section 5, and section 6. We call attention to the figure at the end of the paper, which contains most of our results and gives a survey of the concepts examined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Day,A characterisation of modularity for congruence lattices of algebras, Canad. Math. Bull.12 (1969), no. 2., 167–173.
E. Fried, G. Grätzer andR. W. Quackenbush,Uniform Congruence Schemes, Algebra Universalis10 (1980) 176–188.
E. Fried andA. F. Pixley,The dual discriminator function in universal algebra, Acta Sci. Math.41 (1979) 83–100.
G. Grätzer,Universal Algebra, Van Nostrand, Princeton, 1968.
B. Jónsson,Algebras whose congruence lattices are distributive, Math. Scand.21 (1967), 110–121.
A. I. Mal'cev,On the general theory of algebraic systems, Mat. Sb. (77)35 (1954) 3–20.
A. F. Pixley,Distributivity and Permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc.14 (1963), no. 1. 105–109.
A. F. Pixley,The ternary discriminator function in universal algebra, Math. Ann.191 (1971), 167–180.
R. W. Quackenbush,Semi-simple equational classes with distributive congruence lattices, Ann. Univ. Sci. Budapest Sect. Math.17 (1974), 15–19.
R. Magari,The classification of idealizable varieties (Congruenze Ideali TV.), J. of Algebra26 (1973), 152–165.
R. Magari,Varietá a quozienti filtrali, Ann. Univ. Ferrara, Sez. VII. (N.S.) (1969), 5–20.
H. Werner,Discriminator-algebras, Studien zur algebra und ihre anwendungen, Band 6, Akademie-Verlag, Berlin, 1978.
W.Taylor,Hyperidentities and hypervarieties, to appear in Aequationnes Mathematicae.
Author information
Authors and Affiliations
Additional information
Thanks are also due to Peter Krauss, who called our attention to some gaps in our proofs in a previous version of the present paper.
Rights and permissions
About this article
Cite this article
Fried, E., Kiss, E.W. Connections between congruence-lattices and polynomial properties. Algebra Universalis 17, 227–262 (1983). https://doi.org/10.1007/BF01194534
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01194534