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Trac #17160: Finitely generated axiom for (mutiplicative) magmas, sem…
…igroups, monoids, groups This introduce an axiom FinitelyGeneratedAsMagma, as well as related categories with axioms for magmas, semigroups and groups:: {{{ sage: Groups().FinitelyGeneratedAsMagma() Category of finitely generated groups }}} For ease of notations, when there is no ambiguity, one can do:: {{{ sage: Groups().FinitelyGenerated() Category of finitely generated groups }}} One motivation for this change (for #8678) is that finite semigroups in Sage used to be automatically endowed with an `EnumeratedSets` structure; the default enumeration is then obtained by iteratively multiplying the semigroup generators. This forced any finite semigroup to either implement an enumeration, or provide semigroup generators; this was often inconvenient. Instead, finite semigroups that provide a distinguished finite set of generators with `semigroup_generators` should now explicitly declare themselves in the category of `FinitelyGeneratedSemigroups`: {{{ sage: Semigroups().FinitelyGenerated() Category of finitely generated semigroups }}} This is a backward incompatible change. TODO: - Use the occasion to migrate TransitiveIdeal to RecursivelyEnumeratedSet URL: http://trac.sagemath.org/17160 Reported by: nthiery Ticket author(s): Nicolas M. Thiéry Reviewer(s): Travis Scrimshaw
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