MTL-algebras as rotations of basic hoops

2019 ◽  
Vol 29 (5) ◽  
pp. 763-784 ◽  
Author(s):  
Paolo Aglianò ◽  
Sara Ugolini
Keyword(s):  
Amalgamation Property ◽  
Lattice Of Subvarieties ◽  
Wajsberg Hoops ◽  
Splitting Algebras

Abstract In this paper, we use the generalize d rotation construction to lift results from the lattice of subvarieties of basic hoops to some parts of the lattice of subvarieties of monoidal t-norm based logic-algebras. In particular, we study splitting algebras for (the lattice of subvarieties of) varieties generated by generalized rotations of basic hoops and relevant subvarieties such as Wajsberg hoops, cancellative hoops and Gödel hoops. Finally, we show that the generalized rotation construction preserves the amalgamation property.

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.

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.

A Note on Saturated Models for Many-Valued Logics

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Tommaso Flaminio ◽  
Matteo Bianchi
Keyword(s):  
Representation Theorem ◽  
Amalgamation Property ◽  
Short Paper ◽  
Fuzzy Logics ◽  
Saturated Models ◽  
Joint Embedding ◽  
Joint Embedding Property

AbstractIn this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.

scholarly journals On some varieties of ai-semirings satisfying xp+1 ≈ x

2018 ◽  
Vol 16 (1) ◽  
pp. 913-923
Author(s):  
Aifa Wang ◽  
Yong Shao
Keyword(s):  
Distributive Lattice ◽  
Finitely Based ◽  
Finitely Generated ◽  
Lattice Of Subvarieties

AbstractThe aim of this paper is to study the lattice of subvarieties of the ai-semiring variety defined by the additional identities$$\begin{array}{} \displaystyle x^{p+1}\approx x\,\,\mbox{and}\,\,zxyz\approx(zxzyz)^{p}zyxz(zxzyz)^{p}, \end{array} $$wherepis a prime. It is shown that this lattice is a distributive lattice of order 179. Also, each member of this lattice is finitely based and finitely generated.

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.

Varieties of BL-Algebras III: Splitting Algebras

Studia Logica ◽  
2018 ◽  
Vol 107 (6) ◽  
pp. 1235-1259 ◽  
Author(s):  
Paolo Aglianó
Keyword(s):  
Splitting Algebras

scholarly journals Lattices of pseudovarieties

1989 ◽  
Vol 46 (2) ◽  
pp. 177-183 ◽  
Author(s):  
P. Agliano ◽  
J. B. Nation
Keyword(s):  
Finite Lattice ◽  
Finite Variety ◽  
Locally Finite ◽  
Finite Height ◽  
Lattice Of Subvarieties ◽  
Universal Sentence ◽  
Locally Finite Variety

AbstractWe consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all lattice of subvarieties of varieties V(A) ganarated by algebras A ε P.

scholarly journals Some remarks on divisible polyhedral MV-algebras

2020 ◽  
Vol 70 (1) ◽  
pp. 51-60
Author(s):  
Serafina Lapenta
Keyword(s):  
Amalgamation Property ◽  
Finite Products ◽  
Finitely Presented ◽  
Mv Algebras

AbstractBuilding on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.

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