relational system

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 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 $(P,\{{\le }_P\})$, 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 $(A,\{a\})$ 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 $\{0\}$ and $\{1\}$.

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

• ordered algebraic structures, such as ordered groups $(G,\{\cdot {\text{, }}^{-1}\text{, }e\text{, }{\le }_G\})$ and ordered rings $(R,\{+\text{, }-\text{, }\cdot {\text{, }}^{-1}\text{, }0\text{, }{\le }_R\})$ 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 $(D,\{+\text{, }-\text{, }\cdot {\text{, }}^{-1}\text{, }0\text{, }1\text{, }{\le }_F\})$, etc…

• structures that are not relational are complete lattices (http://planetmath.org/CompleteLattice) 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 $F=\mathrm{\varnothing }$ and an algebraic system is a structure where $R=\mathrm{\varnothing }$.

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