Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices

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• 作者：Antoni Torrens
• 关键词：Residuated lattices ; Semisimple and local residuated lattices ; k ; Radical varieties ; Radical term
• 刊名：Studia Logica
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：104
• 期：5
• 页码：849-867
• 全文大小：576 KB
• 刊物类别：Humanities, Social Sciences and Law
• 刊物主题：Philosophy
  Logic
  Mathematical Logic and Foundations
  Computational Linguistics
  
• 出版者：Springer Netherlands
• ISSN：1572-8730
• 卷排序：104

文摘

In any variety of bounded integral residuated lattice-ordered commutative monoids (bounded residuated lattices for short) the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties (WL_k)_k>0 defined in Cignoli and Torrens (Studia Logica 100:1107–1136, 2012 [7]).