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Commutative integral bounded residuated lattices with an added involution. (English) Zbl 1181.03061
A symmetric residuated lattice is an algebra \(A=(A,\vee ,\wedge ,*,\rightarrow ,\sim,1,0)\) such that \((A,\vee ,\wedge ,*,\rightarrow ,1,0)\) is a commutative integral bounded residuated lattice and the equations \({\sim\sim x}=x\) and \({\sim(x\vee y)}={\sim x}\wedge{\sim y}\) are satisfied.
The aim of this paper is to investigate the properties of the unary operation \(\varepsilon \) defined by \(\varepsilon x={\sim x} \rightarrow 0\).
The authors give necessary and sufficient conditions for \(\varepsilon\) to be an interior operator. Since these conditions are rather restrictive, they consider when an iteration of \(\varepsilon \) is an interior operator. In particular they consider the chain of varieties of symmetric residuated lattices such that the \(n\)-fold iteration of \(\varepsilon\) is a Boolean interior operator. They show that these varieties are semisimple. For \(n=1\), the variety of symmetric Stonean residuated lattices is obtained. Also, the authors characterize the subvarieties admitting representations as subdirect products of chains.

MSC:
03G25 Other algebras related to logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B52 Fuzzy logic; logic of vagueness
06F05 Ordered semigroups and monoids
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