The poset of prime \(l\)-ideals of an abelian \(l\)-group with a strong unit (Q1922900)

An abelian \(\ell\)-group with a strong unit is called a unital \(\ell\)-group. A root system \((X,\leq)\) is said to be spectral if it satisfies the following conditions: (i) Each totally ordered subset of \(X\) has supremum and infimum in \(X\); (ii) if \(x,y\in X\) and \(x<y\), then there are \(x,t\in X\) such that \(x\leq s<t\leq y\), and there is no \(z\in X\) with \(s<z<t\). The main result of the present paper is the following theorem: A poset is order-isomorphic to the poset of prime \(\ell\)-ideals of a unital \(\ell\)-group if and only if it is a spectral root system. The relations between unital \(\ell\)-groups and MV-algebras are essentially applied in the proof of the mentioned theorem.