There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982).[1]
References
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abcPeter Burmeister (1993). "Partial algebras—an introductory survey". In Ivo G. Rosenberg; Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. pp. 1–70.
ISBN978-0-7923-2143-9.