relational systemRelationalSystem2007-01-17 18:13:122009-10-04 20:36:28Definition\usepackage{amssymb,amscd}
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A \emph{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 \PMlinkescapetext{relation on} 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:
\begin{quote}
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 \emph{relation set}. The pair $(A,R)$ is called an $R$-relational system if there is a set $R_A$ such that
\begin{itemize}
\item $R_A$ is a set of finitary relations on $A$, called the \emph{relation set} of $A$, and
\item 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 $r_A$.
\end{itemize}
\end{quote}
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:
\begin{itemize}
\item any algebraic or partial algebraic system.
\item a poset $(P,\lbrace \le_P\rbrace)$, 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 $\wedge$).
\item a pointed set $(A,\lbrace a\rbrace)$ 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).
\item a bounded poset $(P,\le_P,0,1)$ is a relational system. It is a poset, with two unary relations $\lbrace 0\rbrace$ and $\lbrace 1\rbrace$.
\item ordered algebraic structures, such as ordered groups $(G,\lbrace \cdot\mbox{, }^{-1}\mbox{, }e\mbox{, }\le_G\rbrace)$ and ordered rings $(R,\lbrace +\mbox{, }-\mbox{, }\cdot\mbox{, }^{-1}\mbox{, }0\mbox{, }\le_R\rbrace)$ 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.
\item ordered partial algebras such as ordered fields $(D,\lbrace +\mbox{, }-\mbox{, }\cdot\mbox{, }^{-1}\mbox{, }0\mbox{, }1\mbox{, }\le_F\rbrace)$, etc...
\item structures that are not relational are \PMlinkname{complete lattices}{CompleteLattice} and topological spaces, because the operations involved are infinitary.
\end{itemize}
\textbf{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 $R=\varnothing$.
\begin{thebibliography}{7}
\bibitem{gg} G. Gr\"{a}tzer: {\em Universal Algebra}, 2nd Edition, Springer, New York (1978).
\end{thebibliography}