In order to achieve a higher degree of generality, in the technical report

A. Jeffrey, S. Schneider, and F.W. Vaandrager. A comparison of additivity axioms in timed transition systems. Report CS-R9366, CWI, Amsterdam, 1993

the authors proposed to consider an algebraic definition of time domain. Since I like that definition, and I have it used it myself in a couple of papers, allow me to use this post to publicize it.

Define a monoid (X,+, 0) to be:

- left-cancellative iff (x + y = x + z) implies (y = z), and
- anti-symmetric iff (x + y = 0) implies (x = y = 0).

All of the structures mentioned above are, of course, time domains, but so is the set {0}. A time domain is non-trivial if D contains at least two elements. Note that every non-trivial time domain does not have a largest element, and is therefore infinite. Note moreover that + is not required to be commutative, so, for instance, suitable sets of ordinals with ordinal addition form a time domain.

I often find it worthwhile to work with time domains specified with the above degree of generality, and to use properties of specific "concrete" time domains only when they are really needed to obtain certain results. However, maybe this is the axiomatic devil in me talking :-)

Why hasn't the above definition become more popular in the literature on timed process algebras?

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