fix: Replace IsLeftCyclic/IsRightCyclic with unified IsCyclic property

[Copilot]

Replaces separate left/right cyclic properties with a single IsCyclic property that correctly captures the mathematical definition: a magma M is cyclic if ∃m ∈ M such that ⟨m⟩ = M, independent of associativity or sidedness.

Changes

• Removed IsLeftCyclic and IsRightCyclic declarations and implementations
• Added IsCyclic property using submagma generation:

  InstallMethod(IsCyclic, "for a magma", [IsMagma],
      function(M)
          return ForAny(Elements(M), m -> Size(Submagma(M, [m])) = Size(M));
  end);

• Updated MagmaIsomorphismInvariantsMatch to use IsCyclic instead of the two separate properties
• Removed test file test_properties_magma_left_right_cylic.tst
• Added test file test_properties_magma_cyclic.tst with test cases:
  • Non-cyclic magma with multiplication table [[2,1,1,1],[2,1,1,1],[2,1,1,1],[2,1,1,1]]
  • Non-cyclic magma using SmallAntimagma(4,1)

The new implementation checks if any single element generates the entire magma via Submagma, making it applicable to general magmas without assumptions about associativity.

Original prompt

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This section details on the original issue you should resolve

<issue_title>[Bug]: Fix "IsCyclic" method</issue_title>
<issue_description>### What happened?

The IsCyclic function must be defined clearly.

We say antimagma is cyclic if there exists element $m \in M$ such that $M \langle m \rangle$. The associativity does not matter here, more-over, left and right sides does not matter too.

1. Remove IsLeftCyclic
2. Remove IsRightCyclic
3. Add IsCyclic, template:

ForAny(Elements(M), m -> Size(Submagma(M, [m])) = Size(M) )

4. Remove tests too.

Version

0.5.0 (Default)

What browsers are you seeing the problem on?

No response

Relevant log output

Code of Conduct

• I agree to follow this project's Code of Conduct</issue_description>

Comments on the Issue (you are @copilot in this section)

• Fixes [Bug]: Fix "IsCyclic" method #221

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[limakzi]

@copilot Good.
Add a test that this magma with such multiplication table:

[ [ 2, 1, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 1 ] ]

It's generated by set $S = {3, 4}$.
There is no single element that generates it.

Add two tests.

1. Raw test with just multiplication table.
2. Test with SmallAntimagma(4,1).

[Copilot]

@copilot Good.
Add a test that this magma with such multiplication table:

...

Added both tests in tst/test_properties_magma_cyclic.tst:

1. Raw test with the multiplication table [[2,1,1,1],[2,1,1,1],[2,1,1,1],[2,1,1,1]]
2. Test with SmallAntimagma(4,1)

Both tests verify that IsCyclic returns false for this non-cyclic magma. Commit: edd0601

[Copilot]

Pull request overview

This PR replaces separate left/right cyclic properties with a unified IsCyclic property for magmas, implementing the mathematical definition that a magma M is cyclic if there exists an element m ∈ M such that the submagma generated by m equals M.

Changes:

• Removed IsLeftCyclic and IsRightCyclic properties and their implementations, which were based on left/right order
• Added IsCyclic property using submagma generation to check if any element generates the entire magma
• Updated MagmaIsomorphismInvariantsMatch to use IsCyclic instead of the two separate properties

Reviewed changes

Copilot reviewed 4 out of 4 changed files in this pull request and generated 2 comments.

File	Description
lib/properties.gd	Removed declarations for IsLeftCyclic and IsRightCyclic; added IsCyclic declaration with mathematical definition in documentation
lib/properties.gi	Removed implementations of IsLeftCyclic and IsRightCyclic; added IsCyclic implementation using Submagma; updated MagmaIsomorphismInvariantsMatch to use IsCyclic
tst/test_properties_magma_left_right_cylic.tst	Removed old test file that tested left/right cyclic properties against groups
tst/test_properties_magma_cyclic.tst	Added new test file with three test cases: two non-cyclic magmas and one cyclic group

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[olexandr-konovalov]

Thanks:

• the two tests only check the cases when IsCyclic returns false. Can we have a test for true as well?
• what's the motivation for this change? Why not to keep all three notions?
• cyclic semigroups are also called monogenic semigroups - although I don't know which term is more common nowadays. Have you thought of monogenic? In my mind, cyclic seems more restrictive, assuming left-normed or right-normed product, while monogenic would better reflect that some arrangement of brackets may result in generating the whole magma.