Abstract.
An algebra is affine complete iff its polynomial operations are the same as all the operations over its universe that are compatible with all its congruences. A variety is affine complete iff all its algebras are. We prove that every affine complete variety is congruence distributive, and give a useful characterization of all arithmetical, affine complete varieties of countable type. We show that affine complete varieties with finite residual bound have enough injectives. We also construct an example of an affine complete variety without finite residual bound.¶ We prove several results concerning residually finite varieties whose finite algebras are congruence distributive, while leaving open the question whether every such variety must be congruence distributive.
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Received February 28, 1997; accepted in final form December 9, 1997.
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Kaarli, K., McKenzie, R. Affine complete varieties are congruence distributive. Algebra univers. 38, 329–354 (1997). https://doi.org/10.1007/s000120050058
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DOI: https://doi.org/10.1007/s000120050058