# Definition of a hereditary relation

Sassone, V., Nielsen, M. and Winskel, G. (1996) Models for Concurrency: Towards a Classification. 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?

• Hereditary Set – Kaveh Jun 13 '11 at 15:49
• @Kaveh: Your answer is not correct. – Dave Clarke Jun 13 '11 at 20:35
• @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 Jun 14 '11 at 11:23
• @Kaveh: That's debatable – hereditary is a regular English word pertaining to inheritance – $\#$ is inherited along $\le$. – Dave Clarke Jun 14 '11 at 11:36
• @Kaveh: “if you substitute the same relation for both $\leq$ and $\#$” then it is transitivity. “hereditary relation”→“transitivity”→“hereditary set”? Or backwards? – beroal Jun 14 '11 at 14:58

## 1 Answer

In Definition 2.1 of Contextual Petri Nets, Asymmetric Event Structures, and Processes, 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!!

• Sorry, my second question was ambiguous, I rewrote it, look, please. – beroal Jun 14 '11 at 10:52