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Latest version of each Zenodo concept by Logica Artificialis. A row shows only the files on that record, and a GitHub address only when the manuscript names one.

4 repositories

• 10.5281/zenodo.234611402026

  Endomorphism counts from primitive orbits and synchronized block actions

  Martino Lupini · Orderability, Posets, Groups

  For a finite poset P, let R(P) = |End(P)|/|Aut(P)|, where endomorphisms preserve the reflexive order. We prove R(P) ≥ 2^{|P|/(64 log₂ |P|)} for |P| ≥ 4 when every chain has at most two elements and the action on each nonsingleton automorphism orbit is primitive. The same bound holds for a class of imprimitive actions: on each interaction component the action must be the full automorphism group of a rooted tree, acting identically on all the identified orbits. The proof combines simultaneous collapses of indices with maps into principal ideals and filters. We give counting lemmas that retain the whole kernel of a block action, and a quantitative collapse, degree, or orbit-size alternative for normal products of alternating groups with natural orbits. We also construct connected twin-free posets with group S₃ wr Sₜ, constant minimum of the two maximum principal-set sizes, and imprimitive action on every orbit, although every orbit-preserving endomorphism is an automorphism. A restricted three-level extension has lower bound 2^{|P|/(128 (log₂ |P|)²)}. These results give partial answers to the finite-poset automorphism problem of Rival and Rutkowski; the unrestricted question remains beyond the arguments here.

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• 10.5281/zenodo.234584502026

  A countable scattered poset without a strongly maximal chain

  Martino Lupini · Orderability, Posets

  A chain C in a partially ordered set is strongly maximal if |D \ C| ≤ |C \ D| for every chain D. We construct a countably infinite poset with no infinite antichain and no strongly maximal chain. The poset is scattered: none of its chains contains a copy of the rational order. This gives a negative answer, already in the countable scattered case, to the strong-chain question posed by Hollom in 2024 and restated in 2026. The construction replaces each point (x, y, n) of Hollom’s layered poset by a finite chain of length 2^x + 2^y. A finite path exchange restricts the possible chains, after which a further finite exchange always improves them. We also analyze a width-two example of Hollom showing that the chain-extension inclusion used in an earlier claimed positive proof need not hold.

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• 10.5281/zenodo.234527882026

  The small index property for the random poset

  Martino Lupini · Posets, Descriptive set theory, Groups

  Let P be the countable universal homogeneous poset. We prove that every abstract subgroup H ≤ Aut(P) with [Aut(P):H] < 2^{ℵ₀} contains the pointwise stabilizer of a finite subset of P. Thus Aut(P) has the small index property, answering the question recorded as open by Barbina and Macpherson in 2005. No regularity assumption on H is required. The proof equips P with a generic strictly increasing rational-valued rank. Its stabilizer has ample generics, and generic pairs of ranks yield a comeager double coset. We show that the small index property passes from a closed subgroup with such a double coset to the ambient Polish group.

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• 10.5281/zenodo.229325952026

  Lattice-Ordered Groups as Full Automorphism Groups of Posets

  Martino Lupini · Orderability, Posets, Groups

  No abstract on this deposit.

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