Catalogue of min-balanced systems on small sets

This catalogue contains all permutational types of non-trivial min-balanced systems on sets having at most \(5\) elements. We present a number of characteristics of these permutational types. In order to shorten the notation we write \(abc\) instead of \(\{a,b,c\}\).

A few definitions

Given a non-empty finite basic set \(N\) with \(|N|\geq2\), any min-balanced system on \(N\) is given by the list of sets in the system: \(\mathcal{B} = \{B_1,\ldots, B_k\}, k \geq 2\), where \(N=B_1 \cup \ldots \cup B_k\).

More specifically, given a fixed non-empty subset \(M\) of the basic set \((M \subseteq N)\) we say that \(\mathcal{B} \subseteq \mathcal{P}(N) := \{ S: S \subseteq N\}\) is min-balanced (= minimal balanced) on the set \(M\) if it is a minimal set system such that strictly positive real coefficients \(\lambda_S > 0, S \in \mathcal{B}\) exist such that \(\chi_M = \sum_{S \in \mathcal{B}}\lambda_S\cdot\chi_S\), where \(\chi_S\) denotes the incidence vector of \(S \subseteq N\) (in \(\mathbb{R}^N\)). These coefficients are uniquely determined rational numbers; therefore a unique positive integer \(k \in \mathbb{Z}\) exists such that \(\{k\cdot \lambda_S:S\in\mathcal{B}\}\) are integers with no common prime divisor. The (min-balanced) system \(\mathcal{B}\) is ascribed a standardized inequality $$ \alpha(N)\cdot m(N) + \sum_{S \in \mathcal{B}} \alpha(S)\cdot m(S) + \alpha(\emptyset)\cdot m(\emptyset) \geq 0$$ for a real set function \(m: \mathcal{P}(N) \rightarrow \mathbb{R}\), where \(\mathcal{P}(N)\) is the power set of \(N\). Specifically, \(\alpha(N) = k\), \(\alpha(S) = -k\cdot\lambda_S\) for \(S\in\mathcal{B}\) and \(\alpha(\emptyset) = -\alpha(N)-\sum_{S\in\mathcal{B}}\alpha(S)\). These standardized coefficients \(\alpha(S)\) have to be integers.

A schematic description of the standardized inequality says which integers occur among the standardized coefficients. It has a form of a formal equality; for example, the inequality ascribed to system \(\mathcal{B} = \{ab,ac,ad,bcd\}\) over \(N=\{a,b,c,d\}\) is \(3\cdot m(N) - m(ab) - m(ac)- m(ad) - 2\cdot m(bcd) + 2\cdot m(\emptyset) \geq 0\) and its schematic description is \(3\times 1 + 1\times 2 = 3 + 2\) meaning that \(3\) times \(|\alpha(S)|\) for \(S\in \mathcal{B}\) is \(1\), \(1\) times \(|\alpha(S)|\) for \(S\in \mathcal{B}\) is \(2\), the coefficient \(\alpha(N)\) is \(3\) and the coefficient \(\alpha(\emptyset)\) is \(2\).

Another characteristics of the inequality is the index of density which is the ratio \(\alpha(N)/(\alpha(N) + \alpha(\emptyset))\), or equivalently the reciprocal value \((\sum_{S\in\mathcal{B}}\lambda_S)^{-1}\) for unique balancing coefficients \(\lambda(S), S\in\mathcal{B}\) of the system.

The cardinality vector characterizes the cardinalities of sets in the system: it has the form \([c_1,\ldots,c_{n-1}]\) where \(n=|N|\) and \(c_i\) is the number of sets \(S\) in \(\mathcal{B}\) of cardinality \(i\).

The multiplicity vector is obtained by ordering numbers \(m_i = |\{S\in\mathcal{B}: i \in S\}|\) for \(i \in N\) in an increasing way.

Every min-balanced system \(\mathcal{B}\) on \(N\) defines and equivalence relation \(\sim\) on \(N\): one has \(i\sim j\) for \(i,j \in N\) whenever, for every \(S\in \mathcal{B}\), one has \(i \in S\) if and only if \(j\in S\). The system \(\mathcal{B}\) can then be assigned a kind of archetypal system \(\widetilde {\mathcal{B}}\) on the factor set \(\widetilde {N}\) of the equivalence \(\sim\) (\(\widetilde {N}\) is the set of equivalence classes of \(\sim\)). Specifically, \(\widetilde {\mathcal{B}} := \{\widetilde {S}: S\in \mathcal{B}\}\) with \(\widetilde {S} := \{[i]: i \in S\}\) where \([i]\) denotes the equivalence class of \(\sim\) containing \(i \in N\). We say that a min-balanced system \(\mathcal{B}\) on \(N\) and a min-balanced system \(\mathcal{C}\) on (non-empty set) \(L\) have the same archetype if there is a one-to-one mapping \(\psi\) between (factor sets) \(\widetilde {N}\) and \(\widetilde {L}\) which maps \(\widetilde {\mathcal{B}}\) to \(\widetilde {\mathcal{C}}\), that is, \(\widetilde {\mathcal{C}} = \{\psi(\widetilde {S}): \widetilde S \in \widetilde {\mathcal{B}}\}\).

Given a min-balanced system \(\mathcal{B}\) on \(N\), its complementary system (relative to \(N\)) is the system \(\bar{\mathcal{B}}:= \{N\setminus S: S\in\mathcal{B}\}\) where \(N\setminus S\) denotes the relative complement of \(S\) in \(N\). A basic fact about a min-balanced systems on \(N\) is that \(\bar{\mathcal{B}}\) is also a min-balanced system on \(N\).

A min-balanced system \(\mathcal{B}\) on \(N\) is called reducible if there exists a non-empty strict subset \(A\) of \(N\) (that is, \(\emptyset \neq A \subset N\)), a min-balanced system \(\mathcal{C}\) on \(A\), and a min-balanced system \(\mathcal{D}\) on \(N\) containing \(A\) (that is, \(A\in \mathcal{D}\)) such that \(\mathcal{B} = \mathcal{C} \cup(\mathcal{D}\setminus \{A\})\) and \(\mathcal{C}\setminus \mathcal{D} \neq \emptyset\). A min-balanced system which is not reducible is called irreducible.