Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term

Let BℝL denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety V of BℝL is a unary term t in the language of bounded residuated lattices such that for every A ∈ V, tA, the interpretation of the term o...

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Detalles Bibliográficos
Publicado: 2012
Materias:
Boolean products
Boolean retraction terms
Free algebras
Residuated lattices
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00393215_v100_n6_p1107_Cignoli
http://hdl.handle.net/20.500.12110/paper_00393215_v100_n6_p1107_Cignoli
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00393215_v100_n6_p1107_Cignoli
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spelling paper:paper_00393215_v100_n6_p1107_Cignoli2023-06-08T15:03:27Z Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term Boolean products Boolean retraction terms Free algebras Residuated lattices Let BℝL denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety V of BℝL is a unary term t in the language of bounded residuated lattices such that for every A ∈ V, tA, the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is a variety such that a variety admits the unary term t as a Boolean retraction term if and only if V ⊆ Vt. Moreover, the equation s(x) = t(x) holds in Vs∪Vt. The radical of A ∈ BℝL, with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety V ⊆ Vt is associated a variety Vr formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in V. Each free algebra in such V is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety Vr, also with the same number of free generators. A hierarchy of subvarieties of BℝL admitting Boolean retraction terms is exhibited. © 2012 Springer Science+Business Media Dordrecht. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00393215_v100_n6_p1107_Cignoli http://hdl.handle.net/20.500.12110/paper_00393215_v100_n6_p1107_Cignoli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Boolean products
Boolean retraction terms
Free algebras
Residuated lattices
spellingShingle Boolean products
Boolean retraction terms
Free algebras
Residuated lattices
Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
topic_facet Boolean products
Boolean retraction terms
Free algebras
Residuated lattices
description Let BℝL denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety V of BℝL is a unary term t in the language of bounded residuated lattices such that for every A ∈ V, tA, the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is a variety such that a variety admits the unary term t as a Boolean retraction term if and only if V ⊆ Vt. Moreover, the equation s(x) = t(x) holds in Vs∪Vt. The radical of A ∈ BℝL, with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety V ⊆ Vt is associated a variety Vr formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in V. Each free algebra in such V is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety Vr, also with the same number of free generators. A hierarchy of subvarieties of BℝL admitting Boolean retraction terms is exhibited. © 2012 Springer Science+Business Media Dordrecht.
title Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
title_short Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
title_full Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
title_fullStr Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
title_full_unstemmed Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term
title_sort varieties of commutative integral bounded residuated lattices admitting a boolean retraction term
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00393215_v100_n6_p1107_Cignoli
http://hdl.handle.net/20.500.12110/paper_00393215_v100_n6_p1107_Cignoli
_version_ 1768544401424908288

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