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Keywords:
Congruence distributive variety; congruence modular variety; congruence permutable variety; idempotent endomorphism
Congruence distributive variety; congruence modular variety; congruence permutable variety; idempotent endomorphism
Summary:
In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term $t(x)$) in every member of the given variety. Here, we try to give a unified account of this phenomenon. In particular, we investigate what happens when various congruence properties—like congruence distributivity, congruence permutability or congruence modularity—are not supposed to hold unrestrictedly in any $\mathbf {A}\in \mathcal {V}$, but only for congruence classes of values of the term operation $t^{\mathbf {A}}$.
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