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Petar Markovic: The Pushing Lemma and the Maximality Theorem

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Wednesday, 16. November 2022, 10:10 - 11:40
In this, fourth lecture in the series, we will finally start applying the theory we learned so far onto the compatible relations (subpowers). We will start by clearing up the details about the thin affine edges which I messed up the previous time. Then we will prove the existence of a term which acts as a different projection on different thin edges, as long as the projection works "in the direction of the thin edges". This result applies to compatible relations (subpowers) of a smooth algebra, proving the Pushing Lemma, which "pushes" the tuples in the direction of the thin edges. Next we will digress a little and move the whole theory from a single or finitely many finite smooth Taylor algebras to the pseudovariety they generate. Then we will prove two similar lemmas, one about the relationship between the thick edges and thin edges and the other about the thin edges in an algebra and thin edges in a factor of the same algebra. The second lemma will have several consequences and restatements, the most useful of which is the Maximality Theorem, which links maximal elements in a subpower and in its projection. The maximality I speak about is with respect to the directed graphs given by [some of] the thin edges.
 
The lecture will be in the Riesz lecture room.

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