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arXiv · 1812.00396

Pseudo-loop conditions

Pierre Gillibert·Julius Jonušas·Michael Pinsker
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Abstract

We initiate the systematic study of loop conditions of arbitrary finite width. Each loop condition is a finite set of identities of a particular shape, and satisfaction of these identities in an algebra is characterized by it forcing a constant tuple into certain invariant relations on powers of the algebra. By showing the equivalence of various loop conditions, we are able to provide a new and short proof of the recent celebrated result stating the existence of a weakest non-trivial idempotent strong Mal'cev condition. We then consider pseudo-loop conditions, a modification suitable for oligomorphic algebras, and show the equivalence of various pseudo-loop conditions within this context. This allows us to provide a new and short proof of the fact that the satisfaction of non-trivial identities of height 1 in a closed oligomorphic core implies the satisfaction of a fixed single identity.

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Pierre Gillibert, Julius Jonušas, Michael Pinsker. 2018-12-02. Pseudo-loop conditions. https://doi.org/10.1112/blms.12286

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