Weak representations of an interval algebra are the objects of interest in the Artificial Intelligence.
Let us give some words about the Mathematicians who worked on this subject.
Allen  defined the calculus of time intervals and Ladkin and Maddux  showed the interpretation of the calculus of time intervals, in terms of representations of a par ticular relation algebra, in the sense of Tarski . They proved that there is, up to an isomorphism, a unique countable representation of this algebra.
Ligozat  generalized the calculus of time intervals to a calculus of n-intervals and presented this generalization expressed in terms of relation algebras As.
Defining canonical functors between the category of weak representations of A n and those of A 1 , Ligozat  extended the results obtained by Ladkin .
Finally, it can be seen that the set of (p, q)-positions can be endowed with natural operations which give rise to a family of quasi—canonical hypergroupoids.
KeywordsIdentity Element Initial Segment Relation Algebra Generalize Interval Standard Construction
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