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(Paolo) that is a fair point but the description is not accurate, the approach has been described in the paper, here it is:
The relationships between the mathematical entities and those defined by Collections Ontology - and detailed in the following sections of the paper - can be defined as follow:
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co:Set ⊂ Set
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co:Bag ⊂ Bag
co:Set ∩ co:Bag = ∅
co:List = co:Bag ∩ Sequence
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There are precise pragmatic reasons motivating this design choice. First of all, we chose not to model the mathematical identity function in CO for a specific reason: to allow one to use CO even when modelling scenarios that describe “collections in terms of the constructive boundaries of those plural entities that form themselves a whole". Therefore, it is possible to consider two sets of people, composed exactly by the same people, as two different research groups without contradictions. .... Second, from an implementation standpoint, the data structures managing co:Set and co:Bag are very different.
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In other words, if you define a Bag and you add ordering you get a List. See axioms here: http://www.essepuntato.it/lode/owlapi/http://purl.org/co/#d4e499
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