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Unify the notation and definition of congruences and cocongruences - #438

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ScriptRaccoon merged 3 commits into
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congruence-notation
Oct 27, 2026
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ScriptRaccoon merged 3 commits into
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congruence-notation

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This PR makes the notation for congruences and cocongruences consistent across all YAML files and content pages.

Notation

  • Congruences: a congruence is now always written as $q_1, q_2 : R \rightrightarrows X$, with $R$ for the relation object (previously also $E$, $f, g$, or $p_1, p_2$).
  • Cocongruences: a cocongruence is now always written as $j_1, j_2 : X \rightrightarrows C$, with $C$ for the corelation object (previously also $E$).
  • Exceptions: where $R$ or $C$ already denotes something else, such as a morphism in Rel or the ring in Projfg(R), the relation object keeps a different letter.

Definition as parallel pairs

Some descriptions and proofs treated a congruence as a single morphism $R \hookrightarrow X \times X$ (or a cocongruence as $X \sqcup X \twoheadrightarrow C$). They now all use the parallel pair from the definition in "quotients of congruences" and pass to the induced morphism only when needed.

  • Malcev and co-Malcev: the descriptions now define reflexive relations and coreflexive corelations as parallel pairs with a common section or retraction, and link to the definition of (co)congruences.
  • Proofs: updated accordingly for Grp (Malcev), Grpc, Grpfg, Ban, and Mono.
  • Preadditive categories: the proof of the implication "preadditive ⟹ quotients of congruences" has been rewritten as a complete argument.

@ScriptRaccoon ScriptRaccoon added the data additions and updates to the database label Oct 27, 2026
@ScriptRaccoon
ScriptRaccoon merged commit dd62ec2 into main Oct 27, 2026
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ScriptRaccoon deleted the congruence-notation branch October 27, 2026 10:04
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