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>
