Transitive Groups Ontology

An informal description of the Ontology of the Transitive Groups Knowledge Base developed so far.

Metadata and Semantics

All Transitive Group instances belong to the owl:Class sd:TransitiveGroup with predicates

• sd:hasURL URI - a URL that points to a remote web page in the Database for Number Fields with more information.

The Transformation Process

Naming scheme

• The names in the tables are in the form of “8T42”. We convert this to a name in the form of “Gr8T42” (http://symbolicdata.org/Data/TransitiveGroups/Gr8T43), because there are some RDF problems with names starting with numbers.

Products

• Groups can be represented as different kind of products. We have the following:
  • sd:isDirectProductOf
  • sd:isWreathProductOf
  • sd:isQuotientOfWreathProductOf

Each of them is a wreath product of two groups. We represent that in RDF as, e.g.,

` sd:left sdtg:Gr3T1 ; sd:right sdtg:Gr3T1 .` In a first version we used blank nodes to store that information, but due to scoping issues blank nodes are not well supported by different RDF tools. - (2016-02-21) Hans-Gert Gräbe translated the data to a notion with named nodes as, e.g., `sdtg:Gr3T1_Gr3T1 sd:left sdtg:Gr3T1 ; sd:right sdtg:Gr3T1 .` #### The Transitive Group Fingerprint A list of all predicates with number of occurences can be generated with a SPARQL-Query: `PREFIX sd: `<http://symbolicdata.org/Data/Model#> `select distinct ?p count(?s)` `where { ?s a sd:TransitiveGroup . ?s ?p ?o .}` `order by ?p` We list all properties with some comments: - sd:hasGenerator String - multiple permutations, e.g., "(1,3,9,7)(2,4,8,6)", "(1,6,9,4)(2,3,8,7)(5,10)", "(2,4,6,8,10)" that generate the group - sd:hasName String - e.g., "[52:42]22" - What's the rule for such a naming? - sd:hasOrder Integer - order of the group - sd:hasOrderFactorization String - e.g., "2\^3 \* 5\^2" - sd:hasProperty URI - multiple of sd:abelian, sd:cyclic, sd:even, sd:irreducible, sd:nilpotent, sd:notSolvable, sd:primitive, sd:semiabelian, sd:simple, sd:solvable - At the moment merely an URI. To be extended to skos:Concept. - sd:isDirectProductOf sd:WreathProduct - sd:isQuotientOfWreathProductOf sd:WreathProduct - sd:isWreathProductOf sd:WreathProduct - sd:numberOfFieldsInDatabase Integer - sd:OrderOfCenter Integer - order of the center of the group