Foat F. Sultanbekov
We describe a (3,3)-homogeneous orthomodular posets for some cardinality of their sets of atoms. We examine a state space and a set of two-valued states of such logics. Particular homogeneous OMPs with exactly k pure states (k=1,...,7, 10,11)
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Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. The question is how to introduce the connective implication in such a logic. We show that this is possible when the orthomodular poset in question is of finite height. The main point is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and this operator is everywhere defined. We present here the complete list of axioms for implication orthomodular posets. This enables us to derive …
We show that for every orthomodular poset P of finite height there can be defined two operators forming an adjoint pair with respect to an order-like relation defined on the power set of P. This enables us to introduce the so-called operator residuated poset corresponding to P from which the original orthomodular poset P can be recovered. Moreover, this correspondence is almost one-to-one. We show that this construction of operators can be applied also to so-called weakly orthomodular and dually weakly orthomodular posets. Examples of such posets are included.
Generalized orthomodular posets were introduced recently by D. Fazio, A. Ledda and the first author of the present paper in order to establish a useful tool for studying the logic of quantum mechanics. They investigated structural properties of these posets. In the present paper we study logical and algebraic properties of these posets. In particular, we investigate conditions under which they can be converted into operator residuated structures. Further, we study their representation by means of algebras (directoids) with everywhere defined operations. We prove congruence properties for the c…
Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptio…
The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset ${\mathbf P}$ is completed into a Dedekind-MacNeille completion $\BDM(\mathbf P)$ then the complete lattice $\BDM(\mathbf P)$ becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. More complicated situation is with orthomodular and pseudo-orthomodular posets…
Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators M(x,y) and R(x,y) in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented poset can be considered as a prototype of such an operator residuated poset. As main results we prove that every Boolean poset as well as every pseudo-orthomodular poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular posets are pr…
Quantum implications in orthomodular posets
本文证明了在正交lub-完备偏序集中可引入多种多值量子蕴涵，并揭示了它们满足序性质时与经典或量子正交逻辑的等价关系及肯定前件规则。
Tense logic based on finite orthomodular posets
本文针对量子逻辑缺乏时间维度的问题，在有限正交模偏序集上构造满足经典与非经典逻辑要求的时态算子，研究其动态对性质及对不精确合取和蕴涵的保持性，并给出由时态算子构造时间偏好关系的方法。
We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.
The concept of operator left residuation has been introduced by the authors in a previous paper. Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single …
On the Set-Representable Orthomodular Posets that are Point-Distinguishing
本文研究集合可表示正交模偏序集的点可区分表示，利用推广的 Stone 表示技术构造同构表示，并证明其上所有二值态均为由点决定的 Dirac 态。
Simultaneous extensions of states on quantum logics
本文研究量子逻辑上两个布尔子代数状态的同步延拓问题，提出基于二值状态丰富性的充分条件，并证明该条件并非必要。
Tense logics based on posets
本文在基于偏序集的逻辑框架下研究真值随时间变化的动态逻辑，引入带偏好关系的时间集和时态算子，提出多种处理方法以推动后续研究。
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