Post #1030541
2026-04-09 17:47 UTC
Consider the complete lattice of substructures of an algebraic structure, or just submodules of a module, or even just ideals in a commutative ring. Every element of this lattice is a join of »principal« substructures (generated by one element). Is there anything else that is special about the set of principal substructures (or its individual elements) from an order-theoretic point of view?
Replies (2)
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@fogti@chaos.social 2026-04-09 17:52
@jdw this feels very connected to independence systems (and perhaps matroids, but not necessarily)
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@caten@mathstodon.xyz 2026-04-09 20:36
@jdw Both the subalgebra and congruence lattices of an algebra are algebraic lattices. The relevant order-theoretic notion here is that of a compact element, although this captures finite generation rather than generation by a single element.