scholarly journals Strictly join irreducible varieties of residuated lattices

2021 ◽  
Author(s):  
Paolo Aglianò ◽  
Sara Ugolini
Keyword(s):  
Residuated Lattices ◽  
Lattice Of Subvarieties

Abstract We study (strictly) join irreducible varieties in the lattice of subvarieties of residuated lattices. We explore the connections with well-connected algebras and suitable generalizations, focusing in particular on representable varieties. Moreover, we find weakened notions of Halldén completeness that characterize join irreducibility. We characterize strictly join irreducible varieties of basic hoops and use the generalized rotation construction to find strictly join irreducible varieties in subvarieties of $\mathsf{MTL}$-algebras. We also obtain some general results about linear varieties of residuated lattices, with a particular focus on representable varieties, and a characterization for linear varieties of basic hoops.

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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