# 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.