Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

Cvetko-Vah, Karin; Bukovšek, Damjana Kokol; Košir, Tomaž; Kudryavtseva, Ganna

doi:10.1007/s10474-011-0071-9

Cited by 2 publications

(1 citation statement)

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“…The technique to prove it is inspired by [28] and uses the assumption of being weakly directed (this notion has appeared in [28, 10.1, p. 60] as semi-transitive, but the latter term has been introduced in [26] to designate a different semigroup property recurring in a number of articles, e.g. [4,5,6,7,14,15]). Hence, we say that an action of a semigroup S on a set A is weakly directed if for all a, b ∈ A there are f, g ∈ S and c ∈ A such that (f, c) → a and (g, c) → b.…”

Section: Stronger Reconstruction For Monoids and Clonesmentioning

confidence: 99%

On a stronger reconstruction notion for monoids and clones

Behrisch¹,

Vargas-Garc\'\ia²

2019

Preprint

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Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

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