Boolean Products of MV-Algebras: Hypernormal MV-Algebras

Roberto Cignoli; Antoni Torrens Torrell

doi:10.1006/jmaa.1996.0167

Boolean Products of MV-Algebras: Hypernormal MV-Algebras

Journal of Mathematical Analysis and Applications ◽

10.1006/jmaa.1996.0167 ◽

1996 ◽

Vol 199 (3) ◽

pp. 637-653 ◽

Cited By ~ 18

Author(s):

Roberto Cignoli ◽

Antoni Torrens Torrell

Keyword(s):

Boolean Products ◽

Mv Algebras

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Free algebras in varieties of BL-algebras generated by a BLn-chain

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700014117 ◽

2006 ◽

Vol 80 (3) ◽

pp. 419-439 ◽

Cited By ~ 5

Author(s):

Manuela Busaniche ◽

Roberto Cignoli

Keyword(s):

Arbitrary Number ◽

Subject Classification ◽

Free Algebras ◽

Ordinal Sum ◽

Primary 03G25 ◽

Free Generators ◽

Boolean Products ◽

Stone Spaces ◽

Mv Algebras

AbstractFree algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.

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Semi-simple and complete MV-algebras

Algebra Universalis ◽

10.1007/bf01190751 ◽

1992 ◽

Vol 29 (1) ◽

pp. 1-9 ◽

Cited By ~ 37

Author(s):

L. P. Belluce

Keyword(s):

Mv Algebras

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Residuated Structures and Orthomodular Lattices

Studia Logica ◽

10.1007/s11225-021-09946-1 ◽

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.

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Priestley duality for MV-algebras and beyond

Forum Mathematicum ◽

10.1515/forum-2020-0115 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wesley Fussner ◽

Mai Gehrke ◽

Samuel J. van Gool ◽

Vincenzo Marra

Keyword(s):

Large Class ◽

Order Conditions ◽

Enriched Environment ◽

Priestley Duality ◽

Distributive Lattices ◽

First Order ◽

Dual Spaces ◽

Binary Operations ◽

New Perspective ◽

Mv Algebras

Abstract We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

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