The Ordering of Spec R

Abstract

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

Time-dependent coupled harmonic oscillators: Classical and quantum solutions

“…To obtain the quantum states, they used the LR dynamical method. 5,6 As it is well known the exact wave function of the system is the same as the eigenstate of the invariant operator, except for some time-dependent phase factor. For each oscillator, the wave function is expressed in terms of a c-number quantity (ρ) which is solution of the Milne-Pinney (MP) equation.…”

The Order Structure of the Real Spectrum of Commutative Rings

“…On the other hand, the order structure of the Zariski spectrum Spec A of a (commutative, unitary) ring A was investigated in the 1970s by Lewis and Ohm [14,15]; initial results appear also in [12,Section 15]. These results show that the situation is radically different in this case.…”

“…8]; this fails for Spec R unless the set of predecessors of any maximal element is totally ordered. However, our result, together with that of Hochster just mentioned, gives another proof of (part of) Theorem 4.2 of [15] that a jump-dense, complete tree (i.e., a partial order whose reverse is a root system) is order-isomorphic to Spec R for some ring R (they prove that R can be chosen, in addition, to be a Bézout ring).…”

A Poset with Spectral Scott Topology is a Quasialgebraic Domain

“…This definition is in accordance with the terminology in [2] but not with some earlier papers at the borderline between order theory and theoretical computer science (cf. [9,10]). One can obviously see that (X, ≤) is spectral if and only if there exists an ≤-compatible spectral topology on X.…”