Informally in
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, an algebraic theory is a
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
that uses axioms stated entirely in terms of
equation
In mathematics, an equation is a mathematical 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 ...
s between terms with
free variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
s.
Inequalities and
quantifiers are specifically disallowed.
Sentential logic is the subset of
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
involving only algebraic sentences.
The notion is very close to the notion of
algebraic structure
In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
, which, arguably, may be just a synonym.
Saying that a theory is algebraic is a stronger condition than saying it is
elementary.
Informal interpretation
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 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, a binary operation ...
''a'' × ''b'', a nullary operation 1 (
neutral element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
), and a unary operation ''x'' ↦ ''x''
−1 with the rules of
associativity
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
, 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 (just notation, not necessarily th ...
s
* the theory of
lattices
* the theory of
rings
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, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
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:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
whose
objects are
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s 0, 1, 2,..., and which, for each ''n'', has an ''n''-tuple of
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
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 a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s of ''n'' free variables ''X''
1, ..., ''X''
''n'' with
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
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 (mathematics), 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 prope ...
s.
In an algebraic theory, any morphism ''n'' → ''m'' can be described as ''m'' morphisms of
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
''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 can be offered to a market to satisfy the desire or need of a customer.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
...
, 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. Intuitivel ...
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
*
Algebraic definition
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