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
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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 onetoone 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 lattice^{}, 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 singledout 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 BuekenhoutTits 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 equivalence relation^{} $T$ called type, and the other is a symmetric^{} reflexive relation $\mathrm{\#}$ called incidence, such that if $a\mathrm{\#}b$ and $aTb$, then $a=b$ (incident^{} 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 total orders^{}.

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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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structures^{} that are not relational are complete^{} lattices (http://planetmath.org/CompleteLattice) and topological spaces, because the operations^{} 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 $R=\mathrm{\varnothing}$.
References
 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
Title  relational system 
Canonical name  RelationalSystem 
Date of creation  20130322 16:35:33 
Last modified on  20130322 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 