<p>In Definition 2.1 of <a href="http://www.math.unipd.it/~baldan/Papers/Soft-copy-pdf/aes-ic.pdf" rel="nofollow">Contextual Petri Nets, Asymmetric Event Structures, and Processes</a>, Baldan, Corradini and Montanari are working in the setting of <em>prime event structures</em>, which consist of a set of events $E$ along with binary relations $\#$ and $\le$ on $E$, known as the <em>conflict</em> and <em>causality</em> relations, respectively. This is the close to what is going on in your your setting. </p>

<p>In this context, they state that the relation $\#$ is <em>hereditary with respects to $\le$</em> whenever for all $e_0,e_1,e_2\in E$, if $e_0\#e_1$ and $e_1\le e_2$, then $e_0\#e_2$.</p>

<p>It does not make sense to say that a relation is hereditary on its own; it is always taken with respect to another relation.</p>

<p>In general, saying that a relation is the least hereditary relation is not equivalent to saying that it is the least hereditary, symmetry, irreflexive relation, as the definition of hereditary does not imply that the relation is symmetric.</p>

<p>In the specific case you mention, it does seem to be the case – after going into the details of the paper a little more – that one need not specify that the relation is symmetric or irreflexive, as these come for free from the context. Well spotted!!</p>

CC BY-SA 3.0
2011-06-14T10:52:24.157Z
Sorry, my second question was ambiguous, I rewrote it, look, please.
beroal
CC BY-SA 3.0
2011-06-13T10:51:56.847Z
2011-06-14T11:22:28.300Z
2011-06-14T11:22:28.300Z
Dave Clarke
<p>Sassone, V., Nielsen, M. and Winskel, G. (1996) <a href="http://eprints.ecs.soton.ac.uk/11819/" rel="nofollow">Models for Concurrency: Towards a Classification</a>. Theoretical Computer Science, 170 (1-2). pp. 297-348., p. 307:</p>

<blockquote>
  <p>Given a tree $S$, define … $\#$ is the
  least hereditary, symmetric,
  irreflexive relation on $Tran_S$ such
  that $(s,a,s&#39;)\#(s,b,s&#39;&#39;)$ if $s&#39;\neq s&#39;&#39;$.</p>
</blockquote>

<p>What is a hereditary relation? If I replace in the quote “the least hereditary, symmetric, irreflexive relation” with “the least hereditary relation”, do I get the equivalent statement?</p>

CC BY-SA 3.0
2011-06-13T15:49:03.747Z
[Hereditary Set](http://en.wikipedia.org/wiki/Hereditary_set)
Kaveh
CC BY-SA 3.0
2011-06-13T20:35:35.470Z
@Kaveh: Your answer is not correct.
Dave Clarke
CC BY-SA 3.0
2011-06-14T11:23:43.753Z
@Dave, it isn't an answer, but historically this is where the word is coming. The one in your answer is a generalization, if you substitute the same relation for both $\leq$ and $\#$ (e.g. the set membership relation), then it is the same as the one in the WP article.
Kaveh
CC BY-SA 3.0
2011-06-14T11:36:22.887Z
@Kaveh: That's debatable – hereditary is a regular English word pertaining to inheritance – $\#$ is inherited along $\le$.
Dave Clarke
CC BY-SA 3.0
2011-06-14T14:58:41.457Z
@Kaveh: “if you substitute the same relation for both $\leq$ and $\#$” then it is transitivity. “hereditary relation”→“transitivity”→“hereditary set”? Or backwards?
beroal
CC BY-SA 3.0
2011-06-14T15:21:53.550Z
@beroal, it seems that I have made a mistake there. Not all hereditary sets are transitive. A set is hereditary iff all of its members are hereditary, or equivalently the transitive closure of the set contains only sets. It is not as simple as I thought. ($B$ is hereditary and $A\in B$, then $A$ is hereditary.)
Kaveh
CC BY-SA 3.0
2011-06-14T17:11:53.200Z
@Kaveh, @Dave Clarke: It is hard to track down who first called this property “heredity”. My search in Google Scholar shows that this was a nameless part of the definition of event structures since “Nielsen, Plotkin, Winskel. Petri nets, event structures and domains, part I” 1981, then around 1987-1990 different people start calling it “the principle of conflict heredity”, “conflict is hereditary”.
beroal
CC BY-SA 3.0
2011-06-14T19:48:01.443Z
@beroal: One way of finding out is to email one of the authors of one of these papers, and hope for the best.
Dave Clarke
CC BY-SA 3.0
2011-06-15T07:16:54.693Z
@beroal: It's also considered polite to accept correct answers.
Dave Clarke
<terminology><dc.parallel-comp><concurrency><relational-structures>
Definition of a hereditary relation
beroal
# Definition of a hereditary relation

(Asked by: username_0 on 2011-06-10)

Sassone, V., Nielsen, M. and Winskel, G. (1996) [Models for Concurrency: Towards a Classification](http://eprints.ecs.soton.ac.uk/11819/). Theoretical Computer Science, 170 (1-2). pp. 297-348., p. 307:

> Given a tree $S$, define … $\#$ is the
> least hereditary, symmetric,
> irreflexive relation on $Tran\_S$ such
> that $(s,a,s')\#(s,b,s'')$ if $s'\neq s''$.

What is a hereditary relation? If I replace in the quote “the least hereditary, symmetric, irreflexive relation” with “the least hereditary relation”, do I get the equivalent statement?

**Tags:** terminology, dc.parallel-comp, concurrency, relational-structures

> **Comments:**
> **username_1**: [Hereditary Set](http://en.wikipedia.org/wiki/Hereditary_set)  
> **username_2**: @username_1: Your answer is not correct.  
> **username_1**: @Dave, it isn't an answer, but historically this is where the word is coming. The one in your answer is a generalization, if you substitute the same relation for both $\leq$ and $\#$ (e.g. the set membership relation), then it is the same as the one in the WP article.  
> **username_2**: @username_1: That's debatable – hereditary is a regular English word pertaining to inheritance – $\#$ is inherited along $\le$.  
> **username_0**: @username_1: “if you substitute the same relation for both $\leq$ and $\#$” then it is transitivity. “hereditary relation”→“transitivity”→“hereditary set”? Or backwards?  
> **username_1**: @username_0, it seems that I have made a mistake there. Not all hereditary sets are transitive. A set is hereditary iff all of its members are hereditary, or equivalently the transitive closure of the set contains only sets. It is not as simple as I thought. ($B$ is hereditary and $A\in B$, then $A$ is hereditary.)  
> **username_0**: @username_1, @username_2: It is hard to track down who first called this property “heredity”. My search in Google Scholar shows that this was a nameless part of the definition of event structures since “Nielsen, Plotkin, Winskel. Petri nets, event structures and domains, part I” 1981, then around 1987-1990 different people start calling it “the principle of conflict heredity”, “conflict is hereditary”.  
> **username_2**: @username_0: One way of finding out is to email one of the authors of one of these papers, and hope for the best.  
> **username_2**: @username_0: It's also considered polite to accept correct answers.  

---

## 1 answers

### Answer by username_2 on 2011-06-13. Score: 6

In Definition 2.1 of [Contextual Petri Nets, Asymmetric Event Structures, and Processes](http://www.math.unipd.it/~baldan/Papers/Soft-copy-pdf/aes-ic.pdf), Baldan, Corradini and Montanari are working in the setting of *prime event structures*, which consist of a set of events $E$ along with binary relations $\#$ and $\le$ on $E$, known as the *conflict* and *causality* relations, respectively. This is the close to what is going on in your your setting.

In this context, they state that the relation $\#$ is *hereditary with respects to $\le$* whenever for all $e\_0,e\_1,e\_2\in E$, if $e\_0\#e\_1$ and $e\_1\le e\_2$, then $e\_0\#e\_2$.

It does not make sense to say that a relation is hereditary on its own; it is always taken with respect to another relation.

In general, saying that a relation is the least hereditary relation is not equivalent to saying that it is the least hereditary, symmetry, irreflexive relation, as the definition of hereditary does not imply that the relation is symmetric.

In the specific case you mention, it does seem to be the case – after going into the details of the paper a little more – that one need not specify that the relation is symmetric or irreflexive, as these come for free from the context. Well spotted!!

> **Comments:**
> **username_0**: Sorry, my second question was ambiguous, I rewrote it, look, please.  

---

