A pairingon an arbitrary ground set Ωis a triple P ∶=(U, F, Λ), with U, Λtwo sets and F∶U×Ω →Λa map. Sev-eral properties of pairings arise after considering the Moore set system MPand the abstract simplicial complex NPon Ω, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equiva-lence relation ≈P, respectively called minimaland maximumpartitioners. In the present work we first detect various sufficient condi-tions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal parti-tioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using gener-alized matroidal-like properties of NP. More in detail, we first determine a sufficient condition on Pensuring that the fam-ily NPis a closable finitary simplicial complexand call the resulting pairings attractive. On an arbitrary ground set Ω, at-tractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each X∈MPwith respect to ≈Pall have the same cardinality. Nevertheless, the converse does not hold, neither in the finite case. To this regard, we find some counterexamples inducing us to introduce the class of quasi-attractive pairings. We carried out a detailed analysis of quasi-attractive pairings: for instance we characterize them from a lattice-theoretic point of view and, on a finite ground set Ω, also in term of exchange properties of suitable set systems. Finally, by taking the adjacence matrix of a simple undirected graph Gas a model of pairing, we show that the Petersen graph induces an attractive pairing, while the Erdös’ friend-ship graphsinduce a quasi-attractive, but not attractive, one.
Representation theory for simplicial complexes and matroidal-like properties of minimal partitioners.
Cinzia Bisi;
2025
Abstract
A pairingon an arbitrary ground set Ωis a triple P ∶=(U, F, Λ), with U, Λtwo sets and F∶U×Ω →Λa map. Sev-eral properties of pairings arise after considering the Moore set system MPand the abstract simplicial complex NPon Ω, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equiva-lence relation ≈P, respectively called minimaland maximumpartitioners. In the present work we first detect various sufficient condi-tions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal parti-tioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using gener-alized matroidal-like properties of NP. More in detail, we first determine a sufficient condition on Pensuring that the fam-ily NPis a closable finitary simplicial complexand call the resulting pairings attractive. On an arbitrary ground set Ω, at-tractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each X∈MPwith respect to ≈Pall have the same cardinality. Nevertheless, the converse does not hold, neither in the finite case. To this regard, we find some counterexamples inducing us to introduce the class of quasi-attractive pairings. We carried out a detailed analysis of quasi-attractive pairings: for instance we characterize them from a lattice-theoretic point of view and, on a finite ground set Ω, also in term of exchange properties of suitable set systems. Finally, by taking the adjacence matrix of a simple undirected graph Gas a model of pairing, we show that the Petersen graph induces an attractive pairing, while the Erdös’ friend-ship graphsinduce a quasi-attractive, but not attractive, one.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
