The finite embeddability property for residuated lattices, pocrims and BCK-algebras

2002 ◽  
Vol 48 (3) ◽  
pp. 253-271 ◽  
Author(s):  
W. J. Blok ◽  
C. J. Van Alten
Keyword(s):  
Residuated Lattices ◽  
Finite Embeddability Property

scholarly journals The finite embeddability property for some noncommutative knotted extensions of FL

10.29007/vqt7 ◽  
2018 ◽  
Author(s):  
Riquelmi Cardona
Keyword(s):  
Residuated Lattices ◽  
Finite Model ◽  
Structural Rule ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Embeddability Property ◽  
Commutativity Property ◽  
Weaker Conditions ◽  
Full Algebra

We consider the knotted structural rule x<sup>m</sup>≤x<sup>n</sup> for n different than m and m greater or equal than 1. Previously van Alten proved that commutative residuated lattices that satisfy the knotted rule have the finite embeddability property (FEP). Namely, every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. In our work we replace the commutativity property by some slightly weaker conditions. Particularly, we prove the FEP for the variety of residuated lattices that satisfy the equation xyx=x<sup>2</sup>y and the knotted rule. Furthermore, we investigate some generalizations of this noncommutative property by working with equations that allow us to move variables. We also note that the FEP implies the finite model property. Hence the logics modeled by these residuated lattices are decidable.

The finite embeddability property for noncommutative knotted extensions of RL

2015 ◽  
Vol 25 (03) ◽  
pp. 349-379 ◽  
Author(s):  
R. Cardona ◽  
N. Galatos
Keyword(s):  
Classical Logic ◽  
Proof Theory ◽  
Mathematical Linguistics ◽  
Residuated Lattices ◽  
Automata Theory ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Embeddability Property

The finite embeddability property (FEP) for knotted extensions of residuated lattices holds under the assumption of commutativity, but fails in the general case. We identify weaker forms of the commutativity identity which ensure that the FEP holds. The results have applications outside of order algebra to non-classical logic, establishing the strong finite model property (SFMP) and the decidability for deductions, as well as to mathematical linguistics and automata theory, providing new conditions for recognizability of languages. Our proofs make use of residuated frames, developed in the context of algebraic proof theory.

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.

The subvariety of commutative residuated lattices represented by twist-products

2014 ◽  
Vol 71 (1) ◽  
pp. 5-22 ◽  
Author(s):  
Manuela Busaniche ◽  
Roberto Cignoli
Keyword(s):  
Residuated Lattices

Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices

Studia Logica ◽  
2016 ◽  
Vol 104 (5) ◽  
pp. 849-867
Author(s):  
Antoni Torrens
Keyword(s):  
Residuated Lattices
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