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 $(P,\\{\\leq_{P}\\})$ , where $\\leq_{P}$ 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 $\\leq$ 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 $(P,\\leq_{P},0,1)$ 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 $\\Gamma$ with two binary relations on it: one is an equivalence relation $T$ called type, and the other is a symmetric reflexive relation $\\#$ called incidence, such that if $a\\#b$ and $a T b$ , then $a=b$ (incident objects of the same type are identical).
•
ordered algebraic structures, such as ordered groups $(G,\\{\\cdot\\mbox{, }^{-1}\\mbox{, }e\\mbox{, }\\leq_{G}\\})$ and ordered rings $(R,\\{+\\mbox{, }-\\mbox{, }\\cdot\\mbox{, }^{-1}\\mbox{, }0\\mbox{, }\\leq_{R}\\})$ are also relational systems. They are not algebraic systems because of the additional ordering relations ( $\\leq_{G}$ and $\\leq_{R}$ ) defined on these objects. Note that these orderings are generally considered total orders.
•
ordered partial algebras such as ordered fields $(D,\\{+\\mbox{, }-\\mbox{, }\\cdot\\mbox{, }^{-1}\\mbox{, }0\\mbox{, }1\\mbox{, }\\leq_%{F}\\})$ , 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 $F=\\varnothing$ and an algebraic system is a structure where $R=\\varnothing$ .

References

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

单词	RelationalSystem
释义	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 $(P,\\{\\leq_{P}\\})$ , where $\\leq_{P}$ 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 $\\leq$ 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 $(P,\\leq_{P},0,1)$ 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 $\\Gamma$ with two binary relations on it: one is an equivalence relation $T$ called type, and the other is a symmetric reflexive relation $\\#$ called incidence, such that if $a\\#b$ and $a T b$ , then $a=b$ (incident objects of the same type are identical). • ordered algebraic structures, such as ordered groups $(G,\\{\\cdot\\mbox{, }^{-1}\\mbox{, }e\\mbox{, }\\leq_{G}\\})$ and ordered rings $(R,\\{+\\mbox{, }-\\mbox{, }\\cdot\\mbox{, }^{-1}\\mbox{, }0\\mbox{, }\\leq_{R}\\})$ are also relational systems. They are not algebraic systems because of the additional ordering relations ( $\\leq_{G}$ and $\\leq_{R}$ ) defined on these objects. Note that these orderings are generally considered total orders. • ordered partial algebras such as ordered fields $(D,\\{+\\mbox{, }-\\mbox{, }\\cdot\\mbox{, }^{-1}\\mbox{, }0\\mbox{, }1\\mbox{, }\\leq_%{F}\\})$ , 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 $F=\\varnothing$ and an algebraic system is a structure where $R=\\varnothing$ . References 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
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