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 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:

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:

• any algebraic or partial algebraic system.
• 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 $\wedge$ ).
• 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).
• a bounded poset $(P,\le_P,0,1)$ is a relational system. It is a poset, with two unary relations $\lbrace 0\rbrace$ and $\lbrace 1\rbrace$ .
• 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.
• ordered partial algebras such as ordered fields $(D,\lbrace +\mbox{, }-\mbox{, }\cdot\mbox{, }^{-1}\mbox{, }0\mbox{, }1\mbox{, }\le_F\rbrace)$ , etc...
• structures that are not relational are complete lattices 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 $F=\varnothing$ and an algebraic system is a structure where $R=\varnothing$ .

Bibliography

1

G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).