Informally in

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, an algebraic theory is a

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...

that uses axioms stated entirely in terms of

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

s between terms with

free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

Inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

and quantifiers are specifically disallowed.

Sentential 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 ...

is the subset of

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 ...

involving only algebraic sentences. The notion is very close to the notion of

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 ...

, which, arguably, may be just a synonym. Saying that a theory is algebraic is a stronger condition than saying it is

elementary Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, an ...

Informal interpretation

An algebraic theory consists of a collection of ''n''-ary functional terms with additional rules (axioms). For example, the theory of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

is an algebraic theory because it has three functional terms: a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

''a'' × ''b'', a nullary operation 1 (

neutral element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

), and a unary operation ''x'' ↦ ''x''⁻¹ with the rules of

associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

, neutrality and inverses respectively. Other examples include: * the theory of

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

s * the theory of lattices * the theory of

rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

This is opposed to geometric theory which involves partial functions (or binary relationships) or existential quantors − see e.g.

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

where the existence of points or lines is postulated.

Category-based model-theoretical interpretation

An algebraic theory T is a

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

whose

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

are

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s 0, 1, 2,..., and which, for each ''n'', has an ''n''-tuple of

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s: :''proj_i'': ''n'' → 1, ''i'' = 1, ..., ''n'' This allows interpreting ''n'' as a

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of ''n'' copies of 1. Example: Let's define an algebraic theory T taking hom(''n'', ''m'') to be ''m''-tuples of

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s of ''n'' free variables ''X''₁, ..., ''X''_''n'' with

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

s and with substitution as composition. In this case ''proj_i'' is the same as ''X_i''. This theory ''T'' is called the theory of

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

s. 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 Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

, the

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitive ...

Alg(T, ''E'') of the

category of functors In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...

''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)

References

* Lawvere, F. W., 1963,
Functorial Semantics of Algebraic Theories, Proceedings of the National Academy of Sciences 50, No. 5 (November 1963), 869-872
' * Adámek, J., Rosický, 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 * {{nlab, id=algebraic+theory, title=Algebraic theory Mathematical logic

Informal interpretation

Category-based model-theoretical interpretation

See also

References