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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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