A relational system, loosely speaking, is a pair where is a set and is a set of finitary relations defined on (a finitary relation is just an -ary relation where ; when , it is called a property). Since an -ary operator on a set is an -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 a relation set, if there is a function , the set of natural numbers. For each , call the arity of .

Let be a set and a relation set. The pair is called an -relational system if there is a set such that

• is a set of finitary relations on , called the relation set of , and
• there is a one-to-one correspondence between and , given by , such that the the arity of .

Thus, an algebraic system can be treated as a relational system. A partial algebraic system (or partial algebra for short) is also an relational system. It is defined as , where the each member of the relation set is a function from to , where , and is the arity of . Relations in a partial algebraic system are called partial operators.

Below are some exmamples of relational systems:

• a poset , where is a binary relation, called the partial ordering, on . A lattice, generally considered an algebraic system, can also be considered as a relational system, because it is a poset, and that alone defines the algebraic operations ( and ).
• a pointed set is also a relational system, where a unary relation, or property, is the singled-out element . A pointed set is also an algebraic system, if we treat as the lone nullary operator (constant).
• a bounded poset is a relational system. It is a poset, with two unary relations and .
• ordered algebraic structures, such as ordered groups , , , and ordered rings , , , , 0, are also relational systems. They are not algebraic systems because of the additional ordering relations ( and ) defined on these objects. Note that these orderings are generally considered total orders.
• partial algebras are division rings , , , , 0, , fields (correspond to the same relation set as for a division ring), ordered fields , , , , 0, , , 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 set of function symbols , and a set of relation symbols , so a relational system is a structure where and an algebraic system is a structure where .

Bibliography

1

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