I would be grateful for hints about terminology and/or references
on the concept of "regular presentation" introduced years ago
by Vera Trnkova and myself: it is an equational presentation
such that every equation has the same set of variables on both sides
(possibly with different numbers of occurences). In our book
"Automata and algebras in categories" we mention that every such
a presentation defines a finitary set functor H (whose algebras are
precisely the algebras of the variety) which preserves inverse images.
In a joint paper under preparation the converse is proved: whenever
a finitary set functor preserves inverse images, it has a regular
presentation. Is this known or closely related to known results?

Thanks,
Jiri Adamek

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