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Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic theories were introduced in the Lawvere's 1963 thesis Functorial Semantics of Algebraic Theories; while abstract clones were seemingly first defined in Cohn's 1965 Universal algebra and attributed to Hall. The two therefore appeared in the literature at similar times. Though Cohn mentions that Hall lectured on universal algebra between 1947 – 1951, he does not mention whether abstract clones were defined there. In neither source does Lawvere or Cohn mention the other's work, though it is observed at least as early as Linton's 1966 Some aspects of equational categories that the two are closely related.

Is there any evidence to suggest either definition was inspired by the other, or that these were invented entirely independently?

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    $\begingroup$ The true answer to this question is a matter of the personalities of the people involved and their (lack of) regard for each other. This kind of history will never get written up. Your best bet is to have a private conversation with the senior people that you know. $\endgroup$
    – Paul Taylor
    Commented Jul 1, 2021 at 14:20
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    $\begingroup$ Ideas from clone theory go back to Post's classification of clones on 2-element sets back in the 1940s. It is an anachronism to describe this as clone theory in the modern sense, but it pushes back some of the ideas to well before Cohn. $\endgroup$
    – John Coleman
    Commented Jul 1, 2021 at 14:48
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    $\begingroup$ Note that Post’s classification is actually from 1920 (he worked on it during his PhD), and he announced it at that time, even though the full paper took a long time to get published. This is noted in the introduction of the 1941 paper. $\endgroup$
    – Emil Jeřábek
    Commented Jul 1, 2021 at 15:35
  • $\begingroup$ Wikipedia claims that Hall introduced it in his lectures but their reference is to Cohn.... $\endgroup$
    – Benjamin Steinberg
    Commented Jul 2, 2021 at 0:48

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