Semialgebras and their algebras of differences with partial group actions on them. (Q2868582)

Let \((A,+,\cdot)\) be a unital \(K\)-semialgebra for some additively commutative semiring \((K,+,\cdot)\) with identity. Assume that \(A\) as well as \(K\) are additively cancellative. A partial action of a finite group \(G\) on \(A\) is given by a family \((D_g)_{g\in G}\) of ideals of \(A\) and isomorphisms \(\alpha_g\colon D_{g^{-1}}\to D_g\) such that for all \(g,h\in G\): (i) \(D_1=A\) and \(\alpha_1=\text{id}_A\), (ii) \(\alpha_g(D_h\cap D_{g^{-1}})=D_g\cap D_{gh}\), and (iii) \(\alpha_g\circ\alpha_h=\alpha_{gh}\).NEWLINENEWLINE Assume that every ideal \(D_g\) is subtractive and generated by some central idempotent \(1_d\), and that the intersection \(D=\bigcap D_g\) of all \(D_g\neq 0\) satisfies \(D\neq 0\). Then \(D\) is a subtractive ideal of \(A\) and also a \(K\)-semialgebra with identity \(1_d\). An ideal \(I\) of \(A\) is called \(\alpha\)-invariant if \(\alpha_g(I\cap D_{g^{-1}})=I\cap D_g\) for all \(g\in G\), and \(I\) is called \(\alpha\)-prime if it is \(\alpha\)-invariant and \(BC\subseteq I\) for \(\alpha\)-invariant ideals \(B\) and \(C\) implies \(B\subseteq I\) or \(C\subseteq I\).NEWLINENEWLINE Now, let \(A\) be yoked, i.e.\ for all \(a,b\in A\) there is some \(x\in A\) such that \(a+x=b\) or \(a=b+x\). If \(P\) is a subtractive and \(\alpha\)-prime ideal of \(A\) such that \(1_d\not\in P\), then there exists some \(n\leq |G|\) such that there are exactly \(n\) minimal primes over \(P\) in \(\text{Spec }A\). It is also shown by an example that this is not necessarily true if \(1_d\in P\).