1,721,035 research outputs found
An equational theory for σ -complete orthomodular lattices
No full textThe condition of σ-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of σ-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to σ-complete orthomodular lattices is introduced and a completeness theorem is obtained
Local quantum field logic
No full textAlgebraic quantum field theory provides a rigorous analysis of the structure
of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras
indexed by regions of a Lorentzian manifold. In several cases, this net is represented by
a family of von Neumann algebras, specifically, Type III factors. Local quantum field
logic emerges as a logical system that captures the propositional structure encoded in the
algebras of the net and their respective locality conditions. Specifically, by considering
an expanded language of orthomodular lattices that naturally arises from the Murray
von Neumann dimension theory, we first provide equational conditions in the lattice of
projectors of a von Neumann factor that uniquely characterize the Type III factor within
the Murray-von Neumann classification. This equational system motivates the study of a
variety of algebras with an underlying orthomodular lattice structure, which we shall refer
to as LQF-algebras. A Hilbert-style calculus is also introduced, establishing a completeness
theorem with respect to the variety of LQF-algebras
The Cantor–Bernstein–Schröder theorem via universal algebra
Full text linkThe Cantor–Bernstein–Schröder theorem (CBS-theorem for short) of set theory was generalized by Sikorski and Tarski to σ-complete Boolean algebras. After this, several generalizations of the CBS-theorem, extending the Sikorski–Tarski version to different classes of algebras, have been established. Among these classes there are lattice ordered groups, orthomodular lattices, MV-algebras, residuated lattices, etc. This suggests to consider a common algebraic framework in which the algebraic versions of the CBS-theorem can be formulated. In this work we provide this framework establishing necessary and sufficient conditions for the validity of the theorem. We also show how this abstract framework includes the versions of the CBS-theorem already present in the literature as well as new versions of the theorem extended to other classes such as groups, modules, semigroups, rings, ∗ -rings etc
Logical approach for two-valued states on quantum systems
No full textIn this paper we develop a logical system associated to two-valued states on orthomodular lattices. An completeness theorem with respect to a variety of orthomodular lattices enriched with an unary operation that represents two-valued states is given
Quantum computational structures: Categorical equivalence for square root qMV -algebras
No full textIn this paper we investigate a categorical equivalence between square root
qMV -algebras (a variety of algebras arising from quantum computation) and a category
of preordered semigroups
Pavelka-style completeness in expansions of Łukasiewicz Logic
No full textAn algebraic setting for the validity of Pavelka style completeness for
some natural expansions of Łukasiewicz logic by new connectives and rational constants
is given. This algebraic approach is based on the fact that the standard MValgebra
on the real segment [0, 1] is an injective MV-algebra. In particular the logics
associated with MV-algebras with product and with divisible MV-algebras are considered
Injectives in residuated algebras
No full textInjectives in several classes of structures associated with logic are characterized.
Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BLalgebras,
NM-algebras and bounded hoop
Fuzzy propositional logic associated with quantum computational gates
No full textWe apply residuated structures associated with fuzzy logic to develop certain aspects of 6
information processing in quantum computing from a logical perspective. For this pur- 7
pose, we introduce an axiomatic system whose natural interpretation is the irreversible 8
quantum Poincar ́e structur
Fuzzy approach to quantum Fredkin gate
Full text linkIn the framework of quantum computation with mixed states, we introduce a fuzzy approach to the quantum Fredkin gate. Under this perspective, we investigate the behaviour of the gate applied to factorized and non-factorized quantum states
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