An algebraic theory consists of a collection of n-ary functional terms with additional rules (axioms).
For example, the theory of groups is an algebraic theory because it has three functional terms: a binary operationa × b, a nullary operation 1 (neutral element), and a unary operation x ↦ x−1 with the rules of associativity, neutrality and inverses respectively. Other examples include:
This is opposed to geometric theory which involves partial functions (or binary relationships) or existential quantors − see e.g. Euclidean geometry where the existence of points or lines is postulated.
Example: Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1, ..., Xn with integercoefficients and with substitution as composition. In this case proji is the same as Xi. This theory T is called the theory of commutative rings.
In an algebraic theory, any morphism n → m can be described as m morphisms of signaturen → 1. These latter morphisms are called n-ary operations of the theory.
If E is a category with finite products, the full subcategory Alg(T, E) of the category of functors [T, E] consisting of those functors that preserve finite products is called the category ofT-models or T-algebras.
Note that for the case of operation 2 → 1, the appropriate algebra A will define a morphism
algebraic, theory, informally, mathematical, logic, algebraic, theory, theory, that, uses, axioms, stated, entirely, terms, equations, between, terms, with, free, variables, inequalities, quantifiers, specifically, disallowed, sentential, logic, subset, first,. Informally in mathematical logic an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables Inequalities and quantifiers are specifically disallowed Sentential logic is the subset of first order logic involving only algebraic sentences The notion is very close to the notion of algebraic structure which arguably may be just a synonym Saying that a theory is algebraic is a stronger condition than saying it is elementary Contents 1 Informal interpretation 2 Category based model theoretical interpretation 3 See also 4 ReferencesInformal interpretation editAn algebraic theory consists of a collection of n ary functional terms with additional rules axioms For example the theory of groups is an algebraic theory because it has three functional terms a binary operation a b a nullary operation 1 neutral element and a unary operation x x 1 with the rules of associativity neutrality and inverses respectively Other examples include the theory of semigroups the theory of lattices the theory of ringsThis is opposed to geometric theory which involves partial functions or binary relationships or existential quantors see e g Euclidean geometry where the existence of points or lines is postulated Category based model theoretical interpretation editSee also Lawvere theory and Equational logic An algebraic theory T is a category whose objects are natural numbers 0 1 2 and which for each n has an n tuple of morphisms proji n 1 i 1 nThis allows interpreting n as a cartesian product of n copies of 1 Example Let s define an algebraic theory T taking hom n m to be m tuples of polynomials of n free variables X1 Xn with integer coefficients and with substitution as composition In this case proji is the same as Xi This theory T is called the theory of commutative rings In an algebraic theory any morphism n m can be described as m morphisms of signature n 1 These latter morphisms are called n ary operations of the theory If E is a category with finite products the full subcategory Alg T E of the category of functors T E consisting of those functors that preserve finite products is called the category of T models or T algebras Note that for the case of operation 2 1 the appropriate algebra A will define a morphism A 2 A 1 A 1 A 1 See also editAlgebraic definitionReferences editLawvere F W 1963 Functorial Semantics of Algebraic Theories Proceedings of the National Academy of Sciences 50 No 5 November 1963 869 872 Adamek J Rosicky J Vitale E M Algebraic Theories A Categorical Introduction To General Algebra Kock A Reyes G Doctrines in categorical logic in Handbook of Mathematical Logic ed J Barwise North Holland 1977 Algebraic theory at the nLab Retrieved from https en wikipedia org w index php title Algebraic theory amp oldid 1217808138, wikipedia, wiki, book, books, library,
