Simple, local and subdirectly irreducible state residuated lattices

2021 ◽  
pp. 365-383
Author(s):  
Mohammad Taheri ◽  
Farhad Khaksar Haghani ◽  
Saeed Rasouli
Keyword(s):  
Residuated Lattices ◽  
Subdirectly Irreducible

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

Compatible Operations on Residuated Lattices

Studia Logica ◽  
2011 ◽  
Vol 98 (1-2) ◽  
pp. 203-222 ◽  
Author(s):  
J. L. Castiglioni ◽  
H. J. San Martín
Keyword(s):  
Residuated Lattices

Splittings in varieties of logic

2021 ◽  
pp. 1-48
Author(s):  
Brian A. Davey ◽  
Tomasz Kowalski ◽  
Christopher J. Taylor
Keyword(s):  
Residuated Lattices ◽  
Heyting Algebras ◽  
General Lemma ◽  
Negative Results ◽  
Splitting Algebras

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

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