<p>The correspondence theory of modal logic is concerned with the study of specific types of relations definable by modal formulae. My one sentence description does not do justice to the numerous motivations behind correspondence theory; consider it an explanation of why it relates to your sentence.</p>

<p>Specific examples arising in correspondence theory are the following.</p>

<ol>
<li>The modal formula $\Box p \implies p$ is true in a frame if and only if the frame is reflexive.</li>
<li>The formula $\Box p \implies \Box \Box p$ is true in a frame if and only if the frame is transitive. </li>
</ol>

<p>So the types of relations you are asking for can be viewed as being defined by a finite set of modal formulae. These formulae can in turn be viewed as the axioms of a logic, so your question is directly related to the study of modal logics. You can look up material on the model theory of modal logic for more details.</p>

<ol>
<li><a href="http://www.illc.uva.nl/lgc/translation/papers/BenthemCh2.ps" rel="nofollow">Modal Correspondence Theory</a>, van Benthem</li>
<li><a href="http://rads.stackoverflow.com/amzn/click/0521527147" rel="nofollow">Modal Logic</a>, Blackburn, de Rijke, Venema</li>
<li><a href="http://www.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf" rel="nofollow">Model Theory of Modal Logic</a>, Goranko and Otto</li>
</ol>

CC BY-SA 3.0
2013-04-18T09:33:52.357Z
Thanks Vijay. I know about modal logics, but I was wondering if this kind of relation has been studied somewhere else, and has a name more explicit than K45 frames :-)
BastienMob
CC BY-SA 3.0
2013-04-18T07:53:16.127Z
2013-04-18T07:53:16.127Z
Vijay D
<p>I am currently looking at relations that are both transitive and Euclidean, but not necessarily symmetric, hence not necessarily equivalence relations. I have found nothing about this kind of relations. Does anyone know of some work concerning these relations? And first of all do they have a name? </p>

<p>Thanks!</p>

CC BY-SA 3.0
2013-04-17T22:41:34.800Z
These relations seem pretty simple. Basically they consist of unrelated subclasses of elements, where each subclass consists of an equivalence class, and possibly a collection of unrelated elements "below" that equivalence class in the relation. As a digraph, it would look like a disjoint union of complete bipartite graphs $B_j$, with arcs from a "source" partition $S_j$ to a "target" partition $T_j$, and then connecting all of the nodes in $T_j$ symmetrically, for each $j$.
Niel de Beaudrap
CC BY-SA 3.0
2013-04-17T22:57:16.137Z
Note that in the above description, all elements of each subset $T_j$ would also have self-loops, while none of the elements in any subset $S_j$ would. In particular, such a relation is reflexive iff it is an equivalence relation.
Niel de Beaudrap
CC BY-SA 3.0
2013-04-18T09:31:34.490Z
Thanks for the insight! So in fact you could say that the directed graph of such a relation is the graph of an equivalence relation plus a possibly empty set of nodes with entry degree 0.
BastienMob
CC BY-SA 3.0
2013-04-18T15:38:10.090Z
@BastienMob, could you modify the question to give more context and what you know. It's not obvious that you knew about the modal logic context.
Vijay D
<relational-structures>
Euclidean transitive relations
BastienMob
# Euclidean transitive relations

(Asked by: username_0 on 2013-04-17)

I am currently looking at relations that are both transitive and Euclidean, but not necessarily symmetric, hence not necessarily equivalence relations. I have found nothing about this kind of relations. Does anyone know of some work concerning these relations? And first of all do they have a name?

Thanks!

**Tags:** relational-structures

> **Comments:**
> **username_1**: These relations seem pretty simple. Basically they consist of unrelated subclasses of elements, where each subclass consists of an equivalence class, and possibly a collection of unrelated elements "below" that equivalence class in the relation. As a digraph, it would look like a disjoint union of complete bipartite graphs $B_j$, with arcs from a "source" partition $S_j$ to a "target" partition $T_j$, and then connecting all of the nodes in $T_j$ symmetrically, for each $j$.  
> **username_1**: Note that in the above description, all elements of each subset $T_j$ would also have self-loops, while none of the elements in any subset $S_j$ would. In particular, such a relation is reflexive iff it is an equivalence relation.  
> **username_0**: Thanks for the insight! So in fact you could say that the directed graph of such a relation is the graph of an equivalence relation plus a possibly empty set of nodes with entry degree 0.  
> **username_2**: @username_0, could you modify the question to give more context and what you know. It's not obvious that you knew about the modal logic context.  

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## 1 answers

### Answer by username_2 on 2013-04-18. Score: 1

The correspondence theory of modal logic is concerned with the study of specific types of relations definable by modal formulae. My one sentence description does not do justice to the numerous motivations behind correspondence theory; consider it an explanation of why it relates to your sentence.

Specific examples arising in correspondence theory are the following.

1. The modal formula $\Box p \implies p$ is true in a frame if and only if the frame is reflexive.
2. The formula $\Box p \implies \Box \Box p$ is true in a frame if and only if the frame is transitive.

So the types of relations you are asking for can be viewed as being defined by a finite set of modal formulae. These formulae can in turn be viewed as the axioms of a logic, so your question is directly related to the study of modal logics. You can look up material on the model theory of modal logic for more details.

1. [Modal Correspondence Theory](http://www.illc.uva.nl/lgc/translation/papers/BenthemCh2.ps), van Benthem
2. [Modal Logic](http://rads.stackoverflow.com/amzn/click/0521527147), Blackburn, de Rijke, Venema
3. [Model Theory of Modal Logic](http://www.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf), Goranko and Otto

> **Comments:**
> **username_0**: Thanks Vijay. I know about modal logics, but I was wondering if this kind of relation has been studied somewhere else, and has a name more explicit than K45 frames :-)  

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