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# Smith set

Article Id: WHEBN0000215899
Reproduction Date:

 Title: Smith set Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Smith set

In voting systems, the Smith set, named after John H. Smith, is the smallest non-empty set of candidates in a particular election such that each member defeats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be "Smith-efficient".

A set of candidates where every member of the set pair-wise defeats every member outside of the set is known as a dominating set.

## Contents

• Properties 1
• Schwartz set comparison 2
• Alternative formulation 3
• Algorithms 4
• See also 5
• References 6
• External links 7

## Properties

• The Smith set always exists and is well-defined. There is only one smallest dominating set since dominating sets are nested, non-empty, and the set of candidates is finite.
• The Smith set can have more than one candidate, either because of pair-wise ties or because of cycles, such as in Condorcet's paradox.
• The Condorcet winner, if one exists, is the sole member of the Smith set. If weak Condorcet winners exist, they are in the Smith set.

## Schwartz set comparison

The Schwartz set is closely related to and is always a subset of the Smith set. The Smith set is larger if and only if a candidate in the Schwartz set has a pair-wise tie with a candidate that is not in the Schwartz set.

The Smith set can be constructed from the Schwartz set by repeatedly adding two types of candidates until no more such candidates exist outside the set:

• candidates that have pair-wise ties with candidates in the set,
• candidates that defeat a candidate in the set.

Note that candidates of the second type can only exist after candidates of the first type have been added.

## Alternative formulation

Any binary relation R on a set A can generate a natural partial order on the R-cycle equivalence classes of set A, so that xRy implies [x] ≥ [y].

When R is the Beats-or-Ties binary relation on the set of candidates defined by x Beats-or-Ties y if and only if x pair-wise defeats or ties y, then the resulting partial order is the beat-or-tie order which is a total order. The Smith set is the maximal element of the beat-or-tie order.

## Algorithms

The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ(n3). It can also be calculated using a version of Kosaraju's algorithm or Tarjan's algorithm in time Θ(n2).

## References

• require('Module:No globals')

local function getCatForId( id )

```   local title = mw.title.getCurrentTitle()
local namespace = title.namespace
if namespace == 0 then
```

return ''

```   elseif namespace == 2 and not title.isSubpage then
```

return ''

```   else
```

return ''

```   end
```

end

local function viafLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )
```

end

local function kulturnavLink( id )

```   return '.. id .. ' id'
```

end

local function sikartLink( id )

```   return '.. id .. '&lng=en ' .. id .. ''
```

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function ciniiLink( id )

```   return '.. id .. '?l=en ' .. id .. ''
```

end

local function bneLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function uscongressLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function narapersonLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function naraorganizationLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function mgpLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. ''
```

end

local function rslLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

```   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?\$' ) and
not string.match( id, '^C/0/%d%d?\$' ) and
not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function sbnLink( id )

```   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d\$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )
```

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

local function nclLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

local function sudocLink( id )

```   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function hlsLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.php ' .. id .. ''
```

end

local function lirLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.450.0.html ' .. id .. ''
```

end

local function splitLccn( id )

```   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?\$' ) then
id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)\$', '%1/%2/%3' )
end
if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+\$' ) then
return mw.text.split( id, '/' )
end
return false
```

end

local function append(str, c, length)

```   while str:len() < length do
str = c .. str
end
return str
```

end

local function lccnLink( id )

```   local parts = splitLccn( id )
if not parts then
return false
end
local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )
```

end

local function mbLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )
```

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

```   local total = 0
for i = 1, 15 do
local digit = isni:byte( i ) - 48 --Get integer value
total = (total + digit) * 2
end
local remainder = total % 11
local result = (12 - remainder) % 11
if result == 10 then
return "X"
end
return tostring( result )
```

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

```   id = id:gsub( '[ %-]', ):upper()
if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]\$' ) then
return false
end
if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
return false
end
return id
```

end

local function isniLink( id )

```   id = validateIsni( id )
if not id then
return false
end
return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )
```

end

local function orcidLink( id )

```   id = validateIsni( id )
if not id then
return false
end
id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )
```

end

local function gndLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )
```

end

local function selibrLink( id ) if not string.match( id, '^%d+\$' ) then

```       return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )
```

end

local function bnfLink( id )

```   --Add cb prefix if it has been removed
if not string.match( id, '^cb.+\$' ) then
id = 'cb' .. id
end
```
```   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )
```

end

local function bpnLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )
```

end

local function ridLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )
```

end

local function bibsysLink( id )

```   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )
```

end

local function ulanLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )
```

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

```   local ids = {}
if not item.claims[property] then
return ids
end
for _, statement in pairs( item.claims[property] ) do
if statement.mainsnak.datavalue then
table.insert( ids, statement.mainsnak.datavalue.value )
end
end
return ids
```

end

local function matchesWikidataRequirements( item, reqs )

```   for _, group in pairs( reqs ) do
local property = 'p' .. group[1]
local qid = group[2]
if item.claims[property] ~= nil then
for _, statement in pairs ( item.claims[property] ) do
if statement.mainsnak.datavalue ~= nil then
if statement.mainsnak.datavalue.value['numeric-id'] == qid then
return true
end
end
end
end
end
return false
```

end

local function createRow( id, label, rawValue, link, withUid )

```   if link then
if withUid then
return '* ' .. label .. ' ' .. link .. '\n'
else
return '* ' .. label .. ' ' .. link .. '\n'
end
else
```

return '* \n'

```   end
```

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

```   { 'VIAF', 'VIAF', 214, viafLink },
{ 'LCCN', 'LCCN', 244, lccnLink },
{ 'ISNI', 'ISNI', 213, isniLink },
{ 'ORCID', 'ORCID', 496, orcidLink },
{ 'GND', 'GND', 227, gndLink },
{ 'SELIBR', 'SELIBR', 906, selibrLink },
{ 'SUDOC', 'SUDOC', 269, sudocLink },
{ 'BNF', 'BNF', 268, bnfLink },
{ 'BPN', 'BPN', 651, bpnLink },
{ 'RID', 'ResearcherID', 1053, ridLink },
{ 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
{ 'ULAN', 'ULAN', 245, ulanLink },
{ 'HDS', 'HDS', 902, hlsLink },
{ 'LIR', 'LIR', 886, lirLink },
{ 'MBA', 'MusicBrainz', 434, mbLink },
{ 'MGP', 'MGP', 549, mgpLink },
{ 'NLA', 'NLA', 409, nlaLink },
{ 'NDL', 'NDL', 349, ndlLink },
{ 'NCL', 'NCL', 1048, nclLink },
{ 'NKC', 'NKC', 691, nkcLink },
{ 'Léonore', 'Léonore', 640, leonoreLink },
{ 'SBN', 'ICCU', 396, sbnLink },
{ 'RLS', 'RLS', 947, rslLink },
{ 'Botanist', '[[Author ci   In an analysis of serial decision making based on majority rule, describes the Smith set and the Schwartz set.
```
• require('Module:No globals')

local function getCatForId( id )

```   local title = mw.title.getCurrentTitle()
local namespace = title.namespace
if namespace == 0 then
```

return ''

```   elseif namespace == 2 and not title.isSubpage then
```

return ''

```   else
```

return ''

```   end
```

end

local function viafLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )
```

end

local function kulturnavLink( id )

```   return '.. id .. ' id'
```

end

local function sikartLink( id )

```   return '.. id .. '&lng=en ' .. id .. ''
```

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function ciniiLink( id )

```   return '.. id .. '?l=en ' .. id .. ''
```

end

local function bneLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function uscongressLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function narapersonLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function naraorganizationLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function mgpLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. ''
```

end

local function rslLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

```   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?\$' ) and
not string.match( id, '^C/0/%d%d?\$' ) and
not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function sbnLink( id )

```   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d\$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )
```

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

local function nclLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

local function sudocLink( id )

```   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function hlsLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.php ' .. id .. ''
```

end

local function lirLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.450.0.html ' .. id .. ''
```

end

local function splitLccn( id )

```   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?\$' ) then
id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)\$', '%1/%2/%3' )
end
if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+\$' ) then
return mw.text.split( id, '/' )
end
return false
```

end

local function append(str, c, length)

```   while str:len() < length do
str = c .. str
end
return str
```

end

local function lccnLink( id )

```   local parts = splitLccn( id )
if not parts then
return false
end
local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )
```

end

local function mbLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )
```

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

```   local total = 0
for i = 1, 15 do
local digit = isni:byte( i ) - 48 --Get integer value
total = (total + digit) * 2
end
local remainder = total % 11
local result = (12 - remainder) % 11
if result == 10 then
return "X"
end
return tostring( result )
```

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

```   id = id:gsub( '[ %-]', ):upper()
if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]\$' ) then
return false
end
if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
return false
end
return id
```

end

local function isniLink( id )

```   id = validateIsni( id )
if not id then
return false
end
return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )
```

end

local function orcidLink( id )

```   id = validateIsni( id )
if not id then
return false
end
id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )
```

end

local function gndLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )
```

end

local function selibrLink( id ) if not string.match( id, '^%d+\$' ) then

```       return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )
```

end

local function bnfLink( id )

```   --Add cb prefix if it has been removed
if not string.match( id, '^cb.+\$' ) then
id = 'cb' .. id
end
```
```   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )
```

end

local function bpnLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )
```

end

local function ridLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )
```

end

local function bibsysLink( id )

```   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )
```

end

local function ulanLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )
```

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

```   local ids = {}
if not item.claims[property] then
return ids
end
for _, statement in pairs( item.claims[property] ) do
if statement.mainsnak.datavalue then
table.insert( ids, statement.mainsnak.datavalue.value )
end
end
return ids
```

end

local function matchesWikidataRequirements( item, reqs )

```   for _, group in pairs( reqs ) do
local property = 'p' .. group[1]
local qid = group[2]
if item.claims[property] ~= nil then
for _, statement in pairs ( item.claims[property] ) do
if statement.mainsnak.datavalue ~= nil then
if statement.mainsnak.datavalue.value['numeric-id'] == qid then
return true
end
end
end
end
end
return false
```

end

local function createRow( id, label, rawValue, link, withUid )

```   if link then
if withUid then
return '* ' .. label .. ' ' .. link .. '\n'
else
return '* ' .. label .. ' ' .. link .. '\n'
end
else
```

return '* \n'

```   end
```

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

```   { 'VIAF', 'VIAF', 214, viafLink },
{ 'LCCN', 'LCCN', 244, lccnLink },
{ 'ISNI', 'ISNI', 213, isniLink },
{ 'ORCID', 'ORCID', 496, orcidLink },
{ 'GND', 'GND', 227, gndLink },
{ 'SELIBR', 'SELIBR', 906, selibrLink },
{ 'SUDOC', 'SUDOC', 269, sudocLink },
{ 'BNF', 'BNF', 268, bnfLink },
{ 'BPN', 'BPN', 651, bpnLink },
{ 'RID', 'ResearcherID', 1053, ridLink },
{ 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
{ 'ULAN', 'ULAN', 245, ulanLink },
{ 'HDS', 'HDS', 902, hlsLink },
{ 'LIR', 'LIR', 886, lirLink },
{ 'MBA', 'MusicBrainz', 434, mbLink },
{ 'MGP', 'MGP', 549, mgpLink },
{ 'NLA', 'NLA', 409, nlaLink },
{ 'NDL', 'NDL', 349, ndlLink },
{ 'NCL', 'NCL', 1048, nclLink },
{ 'NKC', 'NKC', 691, nkcLink },
{ 'Léonore', 'Léonore', 640, leonoreLink },
{ 'SBN', 'ICCU', 396, sbnLink },
{ 'RLS', 'RLS', 947, rslLink },
{ 'Botanist', '[[Author ci   Introduces a version of a generalized Condorcet Criterion that is satisfied when pairwise elections are based on simple majority choice, and for any dominating set, any candidate in the set is collectively preferred to any candidate not in the set.  But Smith does not discuss the idea of a smallest dominating set.
```
• require('Module:No globals')

local function getCatForId( id )

```   local title = mw.title.getCurrentTitle()
local namespace = title.namespace
if namespace == 0 then
```

return ''

```   elseif namespace == 2 and not title.isSubpage then
```

return ''

```   else
```

return ''

```   end
```

end

local function viafLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'VIAF' )
```

end

local function kulturnavLink( id )

```   return '.. id .. ' id'
```

end

local function sikartLink( id )

```   return '.. id .. '&lng=en ' .. id .. ''
```

end

local function tlsLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'WIKI') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function ciniiLink( id )

```   return '.. id .. '?l=en ' .. id .. ''
```

end

local function bneLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function uscongressLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function narapersonLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function naraorganizationLink( id )

```   return '.. id .. ' ' .. id .. ''
```

end

local function botanistLink( id ) local id2 = mw.ustring.gsub(id, '%s', function(s) return mw.uri.encode(s, 'PATH') end)

```   return '.. id2 .. ' ' .. id .. ''
```

end

local function mgpLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. ''
```

end

local function rslLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function leonoreLink( id ) -- Identifiants allant de LH/1/1 à LH/2794/54 (légionnaires) -- Identifiants allant de C/0/1 à C/0/84 (84 légionnaires célèbres) -- Identifiants allant de 19800035/1/1 à 19800035/385/51670 (légionnaires décédés entre 1954 et 1977, et quelques dossiers de légionnaires décédés avant 1954)

```   if not string.match( id, '^LH/%d%d?%d?%d?/%d%d?%d?\$' ) and
not string.match( id, '^C/0/%d%d?\$' ) and
not string.match( id, '^19800035/%d%d?%d?%d?/%d%d?%d?%d?%d?\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function sbnLink( id )

```   if not string.match( id, '^IT\\ICCU\\%d%d%d%d%d%d%d%d%d%d\$' ) and not string.match( id, '^IT\\ICCU\\%u%u[%d%u]%u\\%d%d%d%d%d%d\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SBN' )
```

end

local function nkcLink( id ) return '.. id .. '&CON_LNG=ENG ' .. id .. '' end

local function nclLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '&CON_LNG=ENG ' .. id .. ''
```

end

local function ndlLink( id ) return '.. id .. ' ' .. id .. '' end

local function sudocLink( id )

```   if not string.match( id, '^%d%d%d%d%d%d%d%d[%dxX]\$' ) then
return false
end
return '.. id .. ' ' .. id .. ''
```

end

local function hlsLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.php ' .. id .. ''
```

end

local function lirLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. '.450.0.html ' .. id .. ''
```

end

local function splitLccn( id )

```   if id:match( '^%l%l?%l?%d%d%d%d%d%d%d%d%d?%d?\$' ) then
id = id:gsub( '^(%l+)(%d+)(%d%d%d%d%d%d)\$', '%1/%2/%3' )
end
if id:match( '^%l%l?%l?/%d%d%d?%d?/%d+\$' ) then
return mw.text.split( id, '/' )
end
return false
```

end

local function append(str, c, length)

```   while str:len() < length do
str = c .. str
end
return str
```

end

local function lccnLink( id )

```   local parts = splitLccn( id )
if not parts then
return false
end
local lccnType = parts[1] ~= 'sh' and 'names' or 'subjects'
id = parts[1] .. parts[2] .. append( parts[3], '0', 6 )
return '.. lccnType .. '/' .. id .. ' ' .. id .. '' .. getCatForId( 'LCCN' )
```

end

local function mbLink( id )

```   -- TODO Implement some sanity checking regex
return '.. id .. ' ' .. id .. '' .. getCatForId( 'MusicBrainz' )
```

end

--Returns the ISNI check digit isni must be a string where the 15 first elements are digits local function getIsniCheckDigit( isni )

```   local total = 0
for i = 1, 15 do
local digit = isni:byte( i ) - 48 --Get integer value
total = (total + digit) * 2
end
local remainder = total % 11
local result = (12 - remainder) % 11
if result == 10 then
return "X"
end
return tostring( result )
```

end

--Validate ISNI (and ORCID) and retuns it as a 16 characters string or returns false if it's invalid --See http://support.orcid.org/knowledgebase/articles/116780-structure-of-the-orcid-identifier local function validateIsni( id )

```   id = id:gsub( '[ %-]', ):upper()
if not id:match( '^%d%d%d%d%d%d%d%d%d%d%d%d%d%d%d[%dX]\$' ) then
return false
end
if getIsniCheckDigit( id ) ~= string.char( id:byte( 16 ) ) then
return false
end
return id
```

end

local function isniLink( id )

```   id = validateIsni( id )
if not id then
return false
end
return '.. id .. ' ' .. id:sub( 1, 4 ) .. ' ' .. id:sub( 5, 8 ) .. ' '  .. id:sub( 9, 12 ) .. ' '  .. id:sub( 13, 16 ) .. '' .. getCatForId( 'ISNI' )
```

end

local function orcidLink( id )

```   id = validateIsni( id )
if not id then
return false
end
id = id:sub( 1, 4 ) .. '-' .. id:sub( 5, 8 ) .. '-'  .. id:sub( 9, 12 ) .. '-'  .. id:sub( 13, 16 )
return '.. id .. ' ' .. id .. '' .. getCatForId( 'ORCID' )
```

end

local function gndLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'GND' )
```

end

local function selibrLink( id ) if not string.match( id, '^%d+\$' ) then

```       return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'SELIBR' )
```

end

local function bnfLink( id )

```   --Add cb prefix if it has been removed
if not string.match( id, '^cb.+\$' ) then
id = 'cb' .. id
end
```
```   return '.. id .. ' ' .. id .. ' .. id .. ' (data)' .. getCatForId( 'BNF' )
```

end

local function bpnLink( id )

```   if not string.match( id, '^%d+\$' ) then
return false
end
return '.. id .. ' ' .. id .. '' .. getCatForId( 'BPN' )
```

end

local function ridLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'RID' )
```

end

local function bibsysLink( id )

```   return '.. id .. '&feltselect=bs.autid ' .. id .. '' .. getCatForId( 'BIBSYS' )
```

end

local function ulanLink( id )

```   return '.. id .. ' ' .. id .. '' .. getCatForId( 'ULAN' )
```

end

local function nlaLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'NLA' ) end

local function rkdartistsLink( id ) return '.. id .. ' ' .. id .. '' .. getCatForId( 'RKDartists' ) end

local function getIdsFromWikidata( item, property )

```   local ids = {}
if not item.claims[property] then
return ids
end
for _, statement in pairs( item.claims[property] ) do
if statement.mainsnak.datavalue then
table.insert( ids, statement.mainsnak.datavalue.value )
end
end
return ids
```

end

local function matchesWikidataRequirements( item, reqs )

```   for _, group in pairs( reqs ) do
local property = 'p' .. group[1]
local qid = group[2]
if item.claims[property] ~= nil then
for _, statement in pairs ( item.claims[property] ) do
if statement.mainsnak.datavalue ~= nil then
if statement.mainsnak.datavalue.value['numeric-id'] == qid then
return true
end
end
end
end
end
return false
```

end

local function createRow( id, label, rawValue, link, withUid )

```   if link then
if withUid then
return '* ' .. label .. ' ' .. link .. '\n'
else
return '* ' .. label .. ' ' .. link .. '\n'
end
else
```

return '* \n'

```   end
```

end

--In this order: name of the parameter, label, propertyId in Wikidata, formatting function local conf = {

```   { 'VIAF', 'VIAF', 214, viafLink },
{ 'LCCN', 'LCCN', 244, lccnLink },
{ 'ISNI', 'ISNI', 213, isniLink },
{ 'ORCID', 'ORCID', 496, orcidLink },
{ 'GND', 'GND', 227, gndLink },
{ 'SELIBR', 'SELIBR', 906, selibrLink },
{ 'SUDOC', 'SUDOC', 269, sudocLink },
{ 'BNF', 'BNF', 268, bnfLink },
{ 'BPN', 'BPN', 651, bpnLink },
{ 'RID', 'ResearcherID', 1053, ridLink },
{ 'BIBSYS', 'BIBSYS', 1015, bibsysLink },
{ 'ULAN', 'ULAN', 245, ulanLink },
{ 'HDS', 'HDS', 902, hlsLink },
{ 'LIR', 'LIR', 886, lirLink },
{ 'MBA', 'MusicBrainz', 434, mbLink },
{ 'MGP', 'MGP', 549, mgpLink },
{ 'NLA', 'NLA', 409, nlaLink },
{ 'NDL', 'NDL', 349, ndlLink },
{ 'NCL', 'NCL', 1048, nclLink },
{ 'NKC', 'NKC', 691, nkcLink },
{ 'Léonore', 'Léonore', 640, leonoreLink },
{ 'SBN', 'ICCU', 396, sbnLink },
{ 'RLS', 'RLS', 947, rslLink },
{ 'Botanist', '[[Author ci   Narrows Smith's generalized Condorcet Criterion to the smallest dominating set and calls it Smith's Condorcet Principle.
```
• Discusses the Smith set (named GETCHA) and the Schwartz set (named GOTCHA) as possible standards for optimal collective choice.

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