Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.MinimallyAllowsTerm.FinsetTerms

A charge spectrum $x$ minimally allows a finite set of potential terms $Ts$ if $x$ allows every term $T \in Ts$ and no proper sub-spectrum $y \subsetneq x$ allows all terms in $Ts$. Formally, this is defined such that for every $y$ in the powerset of $x$, $y = x$ if and only if $y$ allows every potential term $T \in Ts$.

For a given charge spectrum $x$ and a finite set of potential terms $Ts$, the property that $x$ minimally allows $Ts$ is decidable. This implies there is a computational procedure to determine if $x$ allows every term $T \in Ts$ while ensuring that no proper subset $y \subsetneq x$ allows all terms in $Ts$.

Let $x$ be a charge spectrum and $Ts$ be a finite set of potential terms. If $x$ minimally allows the set of potential terms $Ts$, then for any potential term $T \in Ts$, the spectrum $x$ allows the term $T$.

For any potential term $T$ and charge spectrum $x$, $x$ minimally allows the finite set of terms $\{T\}$ if and only if $x$ minimally allows the individual term $T$.

Given finite sets $S_5$ and $S_{10}$ of possible charges in a type $\mathcal{Z}$, `minTopBottom S5 S10` is the multiset of all unique charge spectra of the form $(q_{H_d}, q_{H_u}, \{q_{\bar{5}}\}, \{-q_{H_d} - q_{\bar{5}}, q_{10}, q_{H_u} - q_{10}\})$ such that $q_{H_d}, q_{H_u}, q_{\bar{5}} \in S_5$ and $q_{10} \in S_{10}$. This multiset includes every charge spectrum which minimally allows for both top and bottom Yukawa terms.

For any finite sets of charges $S_5$ and $S_{10}$ and any charge spectrum $x$, if $x$ is an element of the multiset $\text{minTopBottom}(S_5, S_{10})$, then $x$ allows a top Yukawa term. Here, $\text{minTopBottom}(S_5, S_{10})$ denotes the multiset of all charge spectra that minimally allow for both top and bottom Yukawa terms, given the sets of possible charges $S_5$ and $S_{10}$.

For any finite sets of charges $S_5$ and $S_{10}$ in a type $\mathcal{Z}$, if a charge spectrum $x$ is an element of the multiset $\text{minTopBottom}(S_5, S_{10})$, then $x$ allows the bottom Yukawa term.

For any $SU(5)$ charge spectrum $x$ and any finite sets of charges $S_{\bar{5}}, S_{10} \subset \mathcal{Z}$, if $x$ minimally allows the set of potential terms $\{ \text{topYukawa}, \text{bottomYukawa} \}$ and $x$ is contained in the set of spectra $\text{ofFinset}(S_{\bar{5}}, S_{10})$, then $x$ is an element of the multiset $\text{minTopBottom}(S_{\bar{5}}, S_{10})$.