nLab
relational structure

Relational structures

Idea

Relational structures are models for a relational theory, that is, a logical theory whose signature is relational.

Definition

A relational structure is a tuple 𝔉\mathfrak{F} whose first component WW, is a non-empty set, and whose remaining components are relations on WW. (We assume that there is at least one relation given.)

In their use in the Kripke semantics of modal logics, the set WW is sometimes called the universe (perhaps better the domain) and the elements of WW are called ‘worlds’, amongst a host of other names! This leads to the terminology ‘possible world semantics’ which is sometimes used.

Examples

  1. poset, (W;)(W;\leq) or more generally a set, WW, with a family of partial orders, { i}\{\leq_i\}, on it;

  2. transition system, (S;{R eeE})(S; \{R_e\mid e\in E\}), (see discussion on the transition system’s page);

  3. a rooted tree can be considered as a relational structure on its set of nodes, by specifying properties of a successor relation;

  4. an arrow structure is a relational structure proposed in modal logic to handle relational versions of groupoids;

  5. a set WW with, on it, an equivalence relation, \sim, or, more generally, a family of equivalence relations, i\sim_i, for ii is some indexing set.

Revised on December 24, 2010 07:52:16 by Toby Bartels (75.88.75.53)