Related papers: Left residuated operators induced by posets with a…
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…
We show that every complemented modular lattice can be converted into a left residuated lattice where the binary operations of multiplication and residuum are term operations. The concept of an operator left residuated poset was introduced…
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…
When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce the connective implication to be everywhere defined and satisfying (left) adjointness with the connective…
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…
Effect algebras and pseudoeffect algebras were introduced by Foulis, Bennett, Dvurecenskij and Vetterlein as so-called quantum structures which serve as an algebraic axiomatization of the logic of quantum mechanics. A natural question…
This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and…
It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators…
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…
As algebraic semantics of the logic of quantum mechanics there are usually used orthomodular posets, i.e. bounded posets with a complementation which is an antitone involution and where the join of orthogonal elements exists and the…
M. Busaniche, R. Cignoli, C. Tsinakis and A. M. Wille showed that every residuated lattice induces a residuation on its full twist product. For their construction they used also lattice operations. We generalize this problem to…
A residuated poset is a structure $\langle A,\le,\cdot,\backslash,/,1 \rangle$ where $\langle A,\le \rangle$ is a poset and $\langle A,\cdot,1 \rangle$ is a monoid such that the residuation law $x\cdot y\le z\iff x\le z/y\iff y\le…
We introduce residuated ortholattices as a generalization of -- and environment for the investigation of -- orthomodular lattices. We establish a number of basic algebraic facts regarding these structures, characterize orthomodular lattices…
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…
Operators play a substantial role in mathematical formalism of quantum mechanics. However, explicit forms of the operators are usually postulated, based on the intuitive assumptions. In this study, variational principle was applied to the…
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…
The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the…
S. Gudder and, later, S. Pulmanova and E. Vincekova, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure…
We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator…
It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation…