Lattice-theoretic properties of algebras of logic

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• 作者：Antonio Ledda^a ; Francesco Paoli^a ; Constantine Tsinakis^b
• 关键词：Primary ; 06F05 ; secondary ; 06D35 ; 06F15 ; 03G10
• 刊名：Journal of Pure and Applied Algebra
• 出版年：October 2014
• 年：2014
• 卷：218
• 期：10
• 页码：1932-1952
• 全文大小：348 K

文摘

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1]$[a,1]$ is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a 唯MV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.