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Partially additive states on orthomodular posets

Josef Tkadlec (1991)

Colloquium Mathematicae

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...

Prime ideal theorem for double Boolean algebras

Léonard Kwuida (2007)

Discussiones Mathematicae - General Algebra and Applications

Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or x ⊲ ∈ F . In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G...

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