Roughness in basic algebras1

2021 ◽  
pp. 1-9
Author(s):  
Jing Wang ◽  
Yichuan Yang
Keyword(s):  
Effect Algebras ◽  
Orthomodular Lattices ◽  
The Relationship ◽  
Mv Algebras

We introduce rough approximations into basic algebras. After investigating elementary properties of the upper (lower) approximations in basic algebras and discussing the convexity of these two approximations in linearly ordered basic algebras, we generalize related results for MV-algebras, lattice ordered effect algebras, and orthomodular lattices to basic algebras. We also study the relationship between upper (lower) rough ideals of basic algebras and upper (lower) approximations of their homomorphic images.

scholarly journals Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

10.14311/1398 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
M. Kalina
Keyword(s):  
Effect Algebra ◽  
Subdirect Product ◽  
Sufficient Conditions ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Orthomodular Lattices ◽  
Atomic Lattice ◽  
Lattice Effect ◽  
Archimedean Lattice ◽  
Mv Algebras

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.

Hull mappings and dimension effect algebras

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Mathematical Models ◽  
Effect Algebra ◽  
Quantum Measurements ◽  
Direct Decomposition ◽  
Dimension Theory ◽  
Effect Algebras ◽  
Orthomodular Lattices ◽  
Physical Systems ◽  
Special Cases ◽  
Mv Algebras

AbstractAn effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

Perfect residuated lattice ordered monoids

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Probability Measure ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Heyting Algebras ◽  
Effect Algebras ◽  
Quantum Logics ◽  
Mv Algebras ◽  
Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

scholarly journals Quantum B-algebras with involutions

2020 ◽  
pp. 2150233
Author(s):  
Lavinia Corina Ciungu
Keyword(s):  
Algebraic Structures ◽  
Wajsberg Hoops ◽  
Residuated Poset ◽  
One To One ◽  
Universal Quantifiers ◽  
The Relationship ◽  
Mv Algebras

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

Quantum B-algebras

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Number Theory ◽  
Algebraic Number Theory ◽  
Connected Components ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Group A ◽  
Ordered Algebras ◽  
Mv Algebras

AbstractThe concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

scholarly journals Rough Operations and Uncertainty Measures on MV-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Maosen Xie
Keyword(s):  
Uncertainty Measures ◽  
The Relationship ◽  
Mv Algebras

We define a lower approximate operation and an upper approximate operation based on a partition on MV-algebras and discuss their properties. We then introduce a belief measure and a plausibility measure on MV-algebras and investigate the relationship between rough operations and uncertainty measures.

scholarly journals Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

scholarly journals Effect algebras with state operator

10.29007/jbdq ◽  
2018 ◽  
Author(s):  
Silvia Pulmannova
Keyword(s):  
Effect Algebra ◽  
Operator Algebras ◽  
Internal State ◽  
Effect Algebras ◽  
State Operator ◽  
Conditional Expectations ◽  
Definition Of ◽  
Mv Algebras

A state operator on effect algebras is introduced as an additive, idempotent and unital mapping from the effect algebra into itself. The definition is inspired by the definition of an internal state on MV-algebras, recently introduced by Flaminio and Montagna. We study state operators on convex effect algebras, and show their relations with conditional expectations on operator algebras.

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