Polyadic algebras (more recently called Halmos algebras) are

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...

s introduced by

Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...

. They are related to

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

analogous to the relationship between

Boolean algebras In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...

and

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

(see

Lindenbaum–Tarski algebra In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory ''T'' consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that ''p'' ...

). There are other ways to relate first-order logic to algebra, including Tarski's

cylindric algebra In mathematics, the notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebra ...

s (when

equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...

is part of the logic) and

Lawvere Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Lawvere studied continuum mechanics as an undergraduate with Clifford Truesd ...

functorial semantics In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

(a categorical approach).

References

Further reading