Characterizations of Posets via Weak States

Weak states on posets are defined which are in some analogy to states on orthomodular posets used in axiomatic quantum mechanics. It is shown how certain properties of the set of weak states characterize certain properties of the underlying poset. Orthomodular posets serve as algebraic models for logics in axiomatic quantum mechanics. States on them are considered which reflect the properties of states of the corresponding physical system. A crucial property of such states is monotonicity. In analogy to these states we define so-called weak states on an arbitrary poset. These weak states are also monotonous and play some role in the characterization of certain algebraic models of quantum systems (cf. [2]). We use properties of the set of weak states in order to characterize certain properties of the underlying poset. In this context semilattices play an important role. For the theory of semilattices we refer the reader to the recent monograph [1]. In the following let P = (P,≤) be an arbitrary but fixed non-empty poset. Definition 1. We call P trivial if |P | = 1. For every positive integer n and a1, . . . , an ∈ P put L(a1, . . . , an) := {x ∈ P |x ≤ a1, . . . , an} and U(a1, . . . , an) := {x ∈ P |x ≥ a1, . . . , an}. P is called connected if its Hasse diagram is a connected graph. P is called upward directed if U(a, b) 6= ∅ for any a, b ∈ P . Now we define weak states on posets. Definition 2. Let f : P → [0, 1]. We call f a 0-weak state on P if both f is monotonous and (f−1({0}),≤) has a greatest element which we denote