On the products of quantum logics
On the rhomboidal heredity in ideal lattices
We show that the class of principal ideals and the class of semiprime ideals are rhomboidal hereditary in the class of modular lattices. Similar results are presented for the class of ideals with forbidden exterior quotients and for the class of prime ideals.
On the set representation of an orthomodular poset
Let P be an orthomodular poset and let B be a Boolean subalgebra of P. A mapping s:P → ⟨0,1⟩ is said to be a centrally additive B-state if it is order preserving, satisfies s(a') = 1 - s(a), is additive on couples that contain a central element, and restricts to a state on B. It is shown that, for any Boolean subalgebra B of P, P has an abundance of two-valued centrally additive B-states. This answers positively a question raised in [13, Open question, p. 13]. As a consequence one obtains a somewhat...
On the structure of numerical event spaces
The probability of the occurrence of an event pertaining to a physical system which is observed in different states determines a function from the set of states of the system to . The function is called a numerical event or multidimensional probability. When appropriately structured, sets of numerical events form so-called algebras of -probabilities. Their main feature is that they are orthomodular partially ordered sets of functions with an inherent full set of states. A classical...
On the tensor product of a Boolean algebra and an orthoalgebra
On the Word Problem for the Modular Lattice with Four Free Generators.
On uniform dimensions of finite groups
Let G be any finite group and L(G) the lattice of all subgroups of G. If L(G) is strongly balanced (globally permutable) then we observe that the uniform dimension and the strong uniform dimension of L(G) are well defined, and we show how to calculate these dimensions.
Opérations dérivées des treillis orthomodulaire (Part 1)
Opérations dérivées des treillis orthomodulaires (Part 2)
Opérations dérivées des treillis orthomodulaires (Part 3)
Operations on graphs determining congruences on graphs
Order properties of splitting subspaces in an inner product space
Ordering of observables and characterization of conditional expectation
Ordres "C.A.C."
Orthocomplemented difference lattices with few generators
The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result...
Orthodox semigroups with complemented congruence lattices.
Orthogonality and complementation in the lattice of subspaces of a finite vector space
We investigate the lattice of subspaces of an -dimensional vector space over a finite field with a prime power together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when is orthomodular. For...
Orthogonality spaces and atomistic orthocomplemented lattices
Orthomodular lattices and closure operations in ordered vector spaces
On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and . It was proved that V is integrally open iff for every orthogonal set A ⊆ V. In this paper we generalize this result. We...
Orthomodular lattices that are horizontal sums of Boolean algebras
The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class of horizontal sums of Boolean algebras, we establish an identity which...