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The study of [[overlapping markup]] uses a similar algebra (see <ref>Steven DeRose. Markup Overlap: A Review and a Horse. In Proceedings of Extreme Markup Languages 2004, Montréal, Québec, August 2-6, 2004.
http://xml.coverpages.org/DeRoseEML2004.pdf</ref>). Its models have more variations depending on whether endpoints of document structures are permitted to be truly co-located, or merely [tangent].
== Temporal statements in the field of cultural heritage ==
In the cultural heritage ontology [[CIDOC CRM]], Allen relations are replaced by so-called ''temporal primitives'', which facilitate the formulation of attestable statements as well as reasoning about these statements.<ref>CIDOC CRM Version 7.3: https://cidoc-crm.org/versions-of-the-cidoc-crm</ref>
Temporal primitives split up the Allen relations into individual statements about the start or end of the intervals. For example, ''X overlaps with Y'' (<math>X \mathbf{\operatorname{o}} Y</math>) can be split as follows:
* <math>X \mathbf{\operatorname{o}} Y</math> ⇔ ''starts before the start of'' (<math>X</math>,<math>Y</math>) ∧ ''ends after the start of'' (<math>X</math>,<math>Y</math>) ∧ ''ends before the end of'' (<math>X</math>,<math>Y</math>)
In addition, the ''equal to'' of the Allen relations is replaced by ''before or with'' and ''after or with''. A simple example:
* The reign of King Harold II ''starts before the start of'' the Battle of Hastings
* The reign/life of Harold II ''ends after or with the start of'' the Battle of Hastings
* The reign/life of Harold II ''ends before or with the end of'' the Battle of Hastings
In the example, it is not necessary to specify whether Harald II was killed at the beginning or during or at the end of the battle, i.e. whether <math>X \mathbf{\operatorname{m}} Y</math>, <math>X \mathbf{\operatorname{o}} Y</math> or <math>X \mathbf{\operatorname{fi}} Y</math> applies. If it is relevant for a specific historical question, it can be specified later.
Another difference is that for temporal, spatial and spatiotemporal statements in CIDOC CRM, the edges (i.e. for temporal statements the start and end) are interpreted as fuzzy. In particular, statements about exact simultaneity are otherwise extremely rare and notoriously difficult to attest.
== Implementations ==
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