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 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 .
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 bounded poset is a relational system. It is a poset, with two unary relations and .
a Buekenhout-Tits geometry can be thought of as a relational system. It consists of a set with two binary relations on it: one is an equivalence relation called type, and the other is a symmetric reflexive relation called incidence, such that if and , then (incident objects of the same type are identical).
ordered partial algebras such as ordered fields , etc…
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 .
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Date of creation||2013-03-22 16:35:33|
|Last modified on||2013-03-22 16:35:33|
|Last modified by||CWoo (3771)|