Universal Algebra Program

Computer Algebra Programs

There are many good computer algebra programs, for instance I have used Axiom as explained on these pages. However Axiom 'hard codes' algebras as program code (algorithms). Here we want algebras to be dynamically created and 'operated on' in various ways, so here I will use a program called UACalc which is available from this site: http://www.uacalc.org/.

So what structure can we use to store a general algebra? Here we use the following elements:

  sub-elements attributes
algebra either: powerAlgebra, productAlgebra, quotientAlgebra, subalgebra, freeAlgebra, bigProductAlgebra, subProductAlgebra or basicAlgebra.  
basicAlgebra algName, desc, cardinality, universe, operations  
productAlgebra algName, desc, cardinality, factors  
powerAlgebra algName, desc, cardinality, powers, root  
superAlgebra algebra  
quotientAlgebra

algName, desc, cardinality, algebra, congruence

 
subAlgebra

algName, desc, cardinality, algebra, subUniverse

 
subProductAlgebra algName, desc, cardinality, generators, universe  
freeAlgebra algName, desc, cardinality, generators, universe  
bigProductAlgebra algName, desc, powers, rootFactors  
algName    
desc    
cardinality    
universe subUniverse  
subUniverse    
operations op  
op

opSymbol,opName,arity,opSymbol,opTable

 
opName    
opTable    
opSymbol    
arity    
elem    
     
root algebra  
factor    
factors    
rootFactors    
powers    
power    
intArray    
stringArray    
congruence    
partition    
generators    
productElem    

Example

<?xml version="1.0"?>
<algebra>
<basicAlgebra>
<algName>m3</algName>
<desc>the diamond</desc>
<cardinality>5</cardinality>
<operations>
<op>
<opSymbol>
<opName>meet</opName>
<arity>2</arity>
</opSymbol>
<opTable>
<intArray>
<row r="[0]">0,0,0,0,0</row>
<row r="[1]">0,1,0,0,1</row>
<row r="[2]">0,0,2,0,2</row>
<row r="[3]">0,0,0,3,3</row>
<row r="[4]">0,1,2,3,4</row>
</intArray>
</opTable>
</op>
<op>
<opSymbol>
<opName>join</opName>
<arity>2</arity>
</opSymbol>
<opTable>
<intArray>
<row r="[0]">0,1,2,3,4</row>
<row r="[1]">1,1,4,4,4</row>
<row r="[2]">2,4,2,4,4</row>
<row r="[3]">3,4,4,3,4</row>
<row r="[4]">4,4,4,4,4</row>
</intArray>
</opTable>
</op>
</operations>
</basicAlgebra>
</algebra>

 

 

 


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