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I have OWL ontology TBOX only (no instances), I need to formally express it using algebra to define some structure. I have looked for that, I found some representations 

(C, P, A) the reference is http://www.dblab.ece.ntua.gr/pubs/uploads/TR-2007-20.pdf

where C is the set of classes, P is the set of properties and A is the set of axioms. A is used to express subsumption, restrictions etc... C includes primitive types used in data properties definition

(C, P, Sub, Func) the reference is http://dl.acm.org/citation.cfm?id=1871946

where C is the set of classes, P is the set of properties, Sub is subsumption relationships and Func relates each class to its applicable properties

Actually, I'm not sure what is the right representation. could you refer my to some reference please if any?

Median Hilal
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    OWL is already specified with a formal semantics that's based on sets; what about that representation *doesn't* work? E.g., section [2.2 Interpretations](http://www.w3.org/TR/owl2-direct-semantics/#Interpretations) of [OWL 2 Web Ontology Language Direct Semantics](http://www.w3.org/TR/owl2-direct-semantics/) begins with "Given a datatype map D and a vocabulary V over D, an interpretation I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) for D and V is a 10-tuple with the following structure: ..." (see original for better formatting). What *doesn't* work about that? – Joshua Taylor Nov 05 '14 at 11:53
  • Actually, I want to keep it: i.) **generic** (avoid details: just classes and properties) and ii.) **technology independent**. (my algebra must fit any ontological conceptualization e.g. OWL, OWL2 or even other ontology languages). I'm not sure whether this makes sense. – Median Hilal Nov 06 '14 at 09:33
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    It doesn't sound much like an *algebra* then. Wouldn't an algebra be something more like where A is a set of *atomic* class symbols, R is a set of *atomic* role/property symbols, and classes non-atomic classes are generated from the constructors {⊔,⊓,¬,∃,∀}? It sounds like you'd just be asking for "description logic" (and while that will handle OWL, OWL2, etc., you can't guarantee something that will handle *all* ontology languages; some have higher expressivity). – Joshua Taylor Nov 06 '14 at 11:58
  • Yes, this is what I mean. Something like (A,R,{⊔,⊓,¬,∃,∀}) but constructs are not even needed, because I don't need to construct new classes/roles, I just need to describe an existing ontology, so all I need is classes, properties and axioms. Thus, the (C, P, A) triple seems suitable. I searched for a formal representation but I found multiple ones. So, I just needed to ensure if this triple is complete (i.e. every ontological entity is even a class or preperty or an axiom). – Median Hilal Nov 10 '14 at 20:54
  • This representation can be further expanded to C = C-domainClasses U C-dataTypes and P = P-object U P-data and A = A-subsumbtion U A-restriction etc... – Median Hilal Nov 10 '14 at 20:55
  • It's not really clear what you're looking for, I think. It sounds like what you want actually *is* description logic. It sounds like you just want to specify your ontology using description logic notation. – Joshua Taylor Nov 10 '14 at 21:05
  • Well, actually I'm working on a research. I'm using ontology as a conceptual model of a specific domain, some algorithms're applied for a graph-like structure of the ontology. An ontology fragment containing required knowledge is the output of these algorithms, this fragment will be transferred into a specific algebra structure using some formal rules. I want to keep the DL details out of the scope of the research level and leave it to the implementation phase. My need for this formal algebra is two folds: i. formalize the extracted ontology fragment, ii. formalize the transformation rules. – Median Hilal Nov 10 '14 at 21:19
  • I intended to use DL first, but I was surprised that no one of the former proposed methods are using it. Most of them use more generic algebra, so this is the reason beside the ones in the previous comment. Though I'm familiar with DL but not a professional, so I may be missing some points :) – Median Hilal Nov 10 '14 at 21:22
  • "no one of the former proposed methods are using it" The question didn't discuss any former proposed methods. What is is that you're trying to represent that the current representations don't handle well? Please clarify the question. – Joshua Taylor Nov 10 '14 at 21:47
  • OK. the (C, P, A) and (C, P, Sub, Func) in the main question are examples, and "former proposed methods" mean those papers in my state-of-the-art, if you like I may refer you to their links :) – Median Hilal Nov 10 '14 at 21:53
  • Why not refer *everyone* and include some references in the question? – Joshua Taylor Nov 11 '14 at 01:32
  • I edit the question and added the reference papers. – Median Hilal Nov 11 '14 at 08:41

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