Inmathematical logic, asignature is a description of thenon-logical symbols of aformal language. Inuniversal algebra, a signature lists the operations that characterize analgebraic structure. Inmodel theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.

Formally, a (single-sorted)signature can be defined as a 4-tuple${\displaystyle \sigma =\left(S_{\mathrm{func}},S_{\mathrm{rel}},S_{\mathrm{const}},\mathrm{ar}\right),}$ where${\displaystyle S_{\mathrm{func}}}$ and${\displaystyle S_{\mathrm{rel}}}$ are disjointsets not containing any other basic logical symbols, called respectively

• function symbols (examples:${\displaystyle +,\times }$),
• relation symbols orpredicates (examples:${\displaystyle \le ,\in }$),
• constant symbols (examples:${\displaystyle 0,1}$),

and a function${\displaystyle \mathrm{ar}:S_{\mathrm{func}}\cup S_{\mathrm{rel}}\to \mathbb{N}}$ which assigns a natural number calledarity to every function or relation symbol. A function or relation symbol is called${\displaystyle n}$-ary if its arity is${\displaystyle n.}$ Some authors define a nullary (${\displaystyle 0}$-ary) function symbol asconstant symbol, otherwise constant symbols are defined separately.

A signature with no function symbols is called arelational signature, and a signature with no relation symbols is called analgebraic signature.^[1]Afinite signature is a signature such that${\displaystyle S_{\mathrm{func}}}$ and${\displaystyle S_{\mathrm{rel}}}$ arefinite. More generally, thecardinality of a signature${\displaystyle \sigma =\left(S_{\mathrm{func}},S_{\mathrm{rel}},S_{\mathrm{const}},\mathrm{ar}\right)}$ is defined as${\displaystyle |\sigma |=\left|S_{\mathrm{func}}\right|+\left|S_{\mathrm{rel}}\right|+\left|S_{\mathrm{const}}\right|.}$

Thelanguage of a signature is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.

In universal algebra the wordtype orsimilarity type is often used as a synonym for "signature". In model theory, a signature${\displaystyle \sigma }$ is often called avocabulary, or identified with the(first-order) language${\displaystyle L}$ to which it provides thenon-logical symbols. However, thecardinality of the language${\displaystyle L}$ will always be infinite; if${\displaystyle \sigma }$ is finite then${\displaystyle |L|}$ will be${\displaystyle {\mathrm{\aleph }}_0}$.

As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:

"The standard signature forabelian groups is${\displaystyle \sigma =(+,-,0),}$ where${\displaystyle -}$ is a unary operator."

Sometimes an algebraic signature is regarded as just a list of arities, as in:

"The similarity type for abelian groups is${\displaystyle \sigma =(2,1,0).}$"

Formally this would define the function symbols of the signature as something like${\displaystyle f_2}$ (which is binary),${\displaystyle f_1}$ (which is unary) and${\displaystyle f_0}$ (which is nullary), but in reality the usual names are used even in connection with this convention.

Inmathematical logic, very often symbols are not allowed to be nullary,^{[citation needed]} so that constant symbols must be treated separately rather than as nullary function symbols. They form a set${\displaystyle S_{\mathrm{const}}}$ disjoint from${\displaystyle S_{\mathrm{func}},}$ on which the arity function${\displaystyle \mathrm{ar}}$ is not defined. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula ofpropositional logic is also a formula offirst-order logic.

An example for an infinite signature uses${\displaystyle S_{\mathrm{func}}=\{+\}\cup \left\{f_a:a\in F\right\}}$ and${\displaystyle S_{\mathrm{rel}}=\{=\}}$ to formalize expressions and equations about avector space over an infinite scalar field${\displaystyle F,}$ where each${\displaystyle f_a}$ denotes the unary operation of scalar multiplication by${\displaystyle a.}$ This way, the signature and the logic can be kept single-sorted, with vectors being the only sort.^[2]

In the context offirst-order logic, the symbols in a signature are also known as thenon-logical symbols, because together with the logical symbols they form the underlying alphabet over which twoformal languages are inductively defined: The set ofterms over the signature and the set of (well-formed)formulas over the signature.

In astructure, aninterpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an${\displaystyle n}$-ary function symbol${\displaystyle f}$ in a structure${\displaystyle \mathbf{A}}$ withdomain${\displaystyle A}$ is a function${\displaystyle f^{\mathbf{A}}:A^n\to A,}$ and the interpretation of an${\displaystyle n}$-ary relation symbol is arelation${\displaystyle R^{\mathbf{A}}\subseteq A^n.}$ Here${\displaystyle A^n=A\times A\times \cdots \times A}$ denotes the${\displaystyle n}$-foldcartesian product of the domain${\displaystyle A}$ with itself, and so${\displaystyle f}$ is in fact an${\displaystyle n}$-ary function, and${\displaystyle R}$ an${\displaystyle n}$-ary relation.

For many-sorted logic and formany-sorted structures, signatures must encode information about the sorts. The most straightforward way of doing this is viasymbol types that play the role of generalized arities.^[3]

Let${\displaystyle S}$ be a set (of sorts) not containing the symbols${\displaystyle \times }$ or${\displaystyle \to .}$

The symbol types over${\displaystyle S}$ are certain words over the alphabet${\displaystyle S\cup \{\times ,\to \}}$: the relational symbol types${\displaystyle s_1\times \cdots \times s_n,}$ and the functional symbol types${\displaystyle s_1\times \cdots \times s_n\to s^{\mathrm{\prime }},}$ for non-negative integers${\displaystyle n}$ and${\displaystyle s_1,s_2,\dots ,s_n,s^{\mathrm{\prime }}\in S.}$ (For${\displaystyle n=0,}$ the expression${\displaystyle s_1\times \cdots \times s_n}$ denotes the empty word.)

A (many-sorted) signature is a triple${\displaystyle (S,P,\mathrm{type})}$ consisting of

• a set${\displaystyle S}$ of sorts,
• a set${\displaystyle P}$ of symbols, and
• a map${\displaystyle \mathrm{type}}$ which associates to every symbol in${\displaystyle P}$ a symbol type over${\displaystyle S.}$

• Term algebra – Freely generated algebraic structure over a given signature

• Burris, Stanley N.; Sankappanavar, H.P. (1981).A Course in Universal Algebra.Springer.ISBN 3-540-90578-2.Free online edition.
• Hodges, Wilfrid (1997).A Shorter Model Theory.Cambridge University Press.ISBN 0-521-58713-1.

• Stanford Encyclopedia of Philosophy: "Model theory"—byWilfred Hodges.
• PlanetMath: Entry "Signature" describes the concept for the case when no sorts are introduced.
• Baillie, Jean, "An Introduction to the Algebraic Specification of Abstract Data Types."

• Mathematical logic
• Model theory
• Universal algebra

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