relational system

A relational systemMathworldPlanetmath, 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 relationMathworldPlanetmathPlanetmathPlanetmath where 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 generalizationPlanetmathPlanetmath 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, the set of natural numbers. For each rR, 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 RA such that

  • RA 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 RA, given by rrA, such that the f(r)= the arity of rA.

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,{P}), where P is a binary relation, called the partial ordering, on P. A latticeMathworldPlanetmathPlanetmath, 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 (A,{a}) is also a relational system, where a unary relation, or property, is the singled-out element aA. A pointed set is also an algebraic system, if we treat a as the lone nullary operator (constant).

  • a bounded poset (P,P,0,1) is a relational system. It is a poset, with two unary relations {0} and {1}.

  • a Buekenhout-Tits geometry can be thought of as a relational system. It consists of a set Γ with two binary relations on it: one is an equivalence relationMathworldPlanetmath T called type, and the other is a symmetricPlanetmathPlanetmath reflexive relation # called incidence, such that if a#b and aTb, then a=b (incidentPlanetmathPlanetmathPlanetmath objects of the same type are identical).

  • ordered algebraic structures, such as ordered groups (G,{-1eG}) and ordered rings (R,{+--10R}) are also relational systems. They are not algebraic systems because of the additional ordering relations (G and R) defined on these objects. Note that these orderings are generally considered total ordersMathworldPlanetmath.

  • ordered partial algebras such as ordered fields (D,{+--101F}), etc…

  • structuresMathworldPlanetmath that are not relational are completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath lattices ( and topological spaces, because the operationsMathworldPlanetmath 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= and an algebraic system is a structure where R=.


Title relational system
Canonical name RelationalSystem
Date of creation 2013-03-22 16:35:33
Last modified on 2013-03-22 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