relational system

A ^{Mathworld Planetmath} , 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 ^{Mathworld Planetmath Planetmath Planetmath} where $n\in N$; 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 ^{Planetmath Planetmath} 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 $f:R\to N$, 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

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  $R_A$ is a set of finitary relations on $A$, called the relation set of $A$, and

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  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:

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  any algebraic or partial algebraic system.

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  a poset $(P,\{{\le }_P\})$, where ${\le }_P$ is a binary relation, called the partial ordering, on $P$. A ^{Mathworld Planetmath Planetmath} , generally considered an algebraic system, can also be considered as a relational system, because it is a poset, and that $\le$ alone defines the algebraic operations ($\vee$ and $\wedge$).

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  a pointed set $(A,\{a\})$ is also a relational system, where a unary relation, or property, is the singled-out element $a\in A$. A pointed set is also an algebraic system, if we treat $a$ as the lone nullary operator (constant).

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  a bounded poset $(P,{\le }_P,0,1)$ is a relational system. It is a poset, with two unary relations $\{0\}$ and $\{1\}$.

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  a Buekenhout-Tits geometry can be thought of as a relational system. It consists of a set $\mathrm{\Gamma }$ with two binary relations on it: one is an ^{Mathworld Planetmath} $T$ called type, and the other is a ^{Planetmath Planetmath} reflexive relation $\mathrm{\#}$ called incidence, such that if $a\mathrm{\#}b$ and $aTb$, then $a=b$ (^{Planetmath Planetmath Planetmath} objects of the same type are identical).

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  ordered algebraic structures, such as ordered groups $(G,\{\cdot {\text{,}}^{-1}\text{,}e\text{,}{\le }_G\})$ and ordered rings $(R,\{+\text{,}-\text{,}\cdot {\text{,}}^{-1}\text{,}0\text{,}{\le }_R\})$ are also relational systems. They are not algebraic systems because of the additional ordering relations (${\le }_G$ and ${\le }_R$) defined on these objects. Note that these orderings are generally considered ^{Mathworld Planetmath} .

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  ordered partial algebras such as ordered fields $(D,\{+\text{,}-\text{,}\cdot {\text{,}}^{-1}\text{,}0\text{,}1\text{,}{\le }_F\})$, etc...

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  ^{Mathworld Planetmath} that are not relational are ^{Planetmath Planetmath Planetmath Planetmath Planetmath Planetmath Planetmath} lattices (http://planetmath.org/CompleteLattice) and topological spaces, because the ^{Mathworld Planetmath} involved are infinitary.

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 $F=\mathrm{\varnothing }$ and an algebraic system is a structure where $R=\mathrm{\varnothing }$.