Polyadic algebra
Encyclopedia
Polyadic algebras are algebraic structure
s 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 algebra
s (when equality is part of the logic) and Lawvere's functorial semantics (categorical
approach).
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s introduced by Paul Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
. They are related to first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 algebra
Cylindric algebra
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. Indeed, cylindric algebras are Boolean algebras equipped with additional...
s (when equality is part of the logic) and Lawvere's functorial semantics (categorical
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
approach).
Further reading
- Paul HalmosPaul HalmosPaul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
, Algebraic Logic, Chelsea Publishing Co. New York (1962)