Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tarski algebra).
There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras (when equality is part of the logic) and Lawvere's functorial semantics (a categorical approach).
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