Relational structure

Builds on Structure Signature Cartesian product

Definition

Let $\sigma$ be a signature with no function symbols and no constant symbols. So each symbol $R \in \sigma$ is a relation symbol with an arity $n_R \ge 1$ . A relational structure of signature $\sigma$ is a structure for $\sigma$ . Explicitly, it is a pair $(A, (R^A)_{R \in \sigma})$ where $A$ is a nonempty set and $R^A \subseteq A^{n_R}$ for each $R \in \sigma$ . Here $A^{n_R}$ is the Cartesian product of $n_R$ copies of $A$ .

Remarks

Example: a simple graph $(V, E)$ , with $E \subseteq V^2$ the set of pairs $(u, v)$ such that $u$ and $v$ are adjacent, is a relational structure with one binary relation.

Non-example: a group $(G, \cdot)$ is not a relational structure, because its signature has a function symbol. It can be turned into one by replacing $\cdot$ with its graph, the ternary relation $\set{(a, b, c) : a \cdot b = c}$ .

Every structure can be encoded in this way as a relational structure. Some authors allow an empty domain for relational structures.

Reading path

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Cartesian product concept
Signature concept
Structure concept
Relational structure definition

Tag	`2E1R3BDDMK`
URI	`m2://2E1R3BDDMK`
Kind	definition
Writing	W0 stub
MSC	08A02, 03C07
Defines	Relational structure
Uses	Structure, Signature, Cartesian product
Changed	2 November 2026

Relational structure

Definition

Remarks

Reading path

Further reading

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