# relational system

A relational system  , loosely speaking, is a pair $(A,R)$ where $A$ is a set and $R$ is a set of finitary relations defined on $A$ (a finitary relation is just an $n$-ary relation    where $n\in\mathbb{N}$; when $n=1$, it is called a property). Since an $n$-ary operator on a set is an $(n+1)$-ary the set, a relational system can be thought of as a generalization  of an algebraic system. We can formalize the notion of a relation system as follows:

Call a set $R$ a relation set, if there is a function $f:R\to\mathbb{N}$, the set of natural numbers. For each $r\in R$, call $f(r)$ the arity of $r$.
Let $A$ be a set and $R$ a relation set. The pair $(A,R)$ is called an $R$-relational system if there is a set $R_{A}$ such that

• $R_{A}$ is a set of finitary relations on $A$, called the relation set of $A$, and

• there is a one-to-one correspondence between $R$ and $R_{A}$, given by $r\mapsto r_{A}$, such that the $f(r)=$ the arity of $r_{A}$.

Since operators and partial operators are special types of relations. algebraic systems and partial algebraic systems can be treated as relational systems.

Below are some exmamples of relational systems:

Remark. Relational systems and algebraic systems are both examples of structures in model theory. Although an algebraic system is a relational system in the sense discussed above, they are treated as distinct entities. A structure involves three objects, a set $A$, a set of function symbols $F$, and a set of relation symbols $R$, so a relational system is a structure where $F=\varnothing$ and an algebraic system is a structure where $R=\varnothing$.

## References

 Title relational system Canonical name RelationalSystem Date of creation 2013-03-22 16:35:33 Last modified on 2013-03-22 16:35:33 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 16 Author CWoo (3771) Entry type Definition Classification msc 08A55 Classification msc 03C07 Classification msc 08A02 Synonym relational structure Related topic AlgebraicSystem Related topic PartialAlgebraicSystem Related topic Structure Related topic StructuresAndSatisfaction