Generalizing Lawvere theories and their functorial semantics to relational theories:
Chad Nester: A Variety Theorem for Relational Universal Algebra, Lecture Notes in Computer Science 13027 (2021) 362-377 [arXiv:2105.04958, doi:10.1007/978-3-030-88701-8_22]
Chad Nester: Partial And Relational Algebraic Theories, PhD thesis, Tallin University (2024) [doi:10.23658/taltech.4/2024]
and to partial algebraic theories:
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