We use Greechie diagrams to construct finite orthomodular lattices "realizable" in the orthomodular lattice of subspaces in a three-dimensional Hilbert space such that the set of two-valued states is not "large" (i.e., full, separating, unital, nonempty, resp.). We discuss the number of elements of such orthomodular lattices, of their sets of (ortho)generators and of their subsets that do not admit a "large" set of two-valued states. We show connections with other results of this type.

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Connections between the weak orthocompleteness and the maximality property in effect algebras are presented. It is proved that an orthomodular poset with the maximality property is disjunctive. A characterization of Archimedean weakly orthocomplete effect algebras is given.Keywords Effect algebra · weakly orthocomplete · maximality property · disjunctive · Archimedean · separable Ovchinnikov [7] introduced weakly orthocomplete orthomodular posets (he called them alternative) as a common generalization of orthocomplete orthomodular posets and orthomodular lattices and showed that they are disjunctive. Weak orthocompleteness is useful in the study of orthoatomisticity and disjunctivity might be used to characterize atomisticity [7,11]. Weak orthocompleteness was generalized by De Simone and Navara [1].Tkadlec [8] introduced the class of orthomodular posets with the maximality property as another common generalization of orthocomplete orthomodular posets and orthomodular lattices. He showed various consequences of this property and generalize it [8,9,10,12].We show that these two notions are incomparable, that maximality property also implies disjunctivity in orthomodular posets and present a characterization of Archimedean weakly orthocomplete effect algebras. We show also some other relations within the class of effect algebras.

We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s : P → [0, 1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P . We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P . Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result. Basic definitions and preliminaries1.1. Definition. An orthomodular poset (abbr. an OMP) is a triple (P, ≤, ′ ) such that 1) (P, ≤) is a partially ordered set with a least element 0 and a greatest element 1, 2) the operation ′ : P → P is an orthocomplementation, i.e. for every a, b ∈ P we have a ′′ = a and b ′ ≤ a ′ whenever a ≤ b, 3) the least upper bound exists for every pair of orthogonal elements inA typical example of an OMP is the lattice of all projections in a Hilbert space or, of course, a Boolean algebra. (We do not assume that P is a lattice. If it is, we call it an orthomodular lattice.)Throughout the paper, P will be an arbitrary OMP and B an arbitrary Boolean subalgebra of P . (By a Boolean subalgebra of P we mean a subset of P which forms a Boolean algebra with respect to ∨ and ′ inherited from P , see also [4], [7].) Let us state our basic definition.1.2. Definition. Let B be a Boolean subalgebra of P . A partially

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