Representation Theorems for Simplicial Complexes and Matroidal-Like Properties of Minimal Partitioners

A {\em pairing} on an arbitrary ground set $\Omega$ is a triple $\mfP:=(U,F,\Lambda)$, with $U,\Lambda$ two sets and $F: U \times \Omega \longrightarrow \Lambda$ a map. Several properties of pairings arise after considering the Moore set system $\mcM\mfp$ and the abstract simplicial complex $\mcN\mfp$ on $\Omega$, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equivalence relation $\approx\mfp$, respectively called {\em minimal} and {\em maximum} partitioners. In the present work we first detect various sufficient conditions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal partitioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using generalized matroidal-like properties of $\mcN\mfp$. More in detail, we first determine a sufficient condition on $\mfP$ ensuring that the family $\mcN\mfp$ is a {\em closable finitary simplicial complex} and call the resulting pairings {\em attractive}. On an arbitrary ground set $\Omega$, attractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each $X \in \mcM\mfp$ with respect to $\approx\mfp$ all 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 {\em 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 $\Omega$, also in term of exchange properties of suitable set systems. Finally, by taking the adjacence matrix of a simple undirected graph $G$ as a model of pairing, we show that the Petersen graph induces an attractive pairing, while the {\em Erd\"{o}s' friendship graphs} induce a quasi-attractive, but not attractive, one.