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On the lattice of \(n\)-filters of an \(\mathrm{LM}_n\)-algebra. (English) Zbl 1129.06010

Let \(L\) be an \(n\)-valued Lukasiewicz-Moisil algebra with endomorphisms \(\varphi_1,\dots,\varphi_n\). The lattice filters of \(L\) that are closed with respect to \(\varphi_1\), and hence with respect to all the endomorphisms \(\varphi_i\), are called \(n\)-filters. It is known that the set \(F_n(L)\) of all \(n\)-filters of \(L\) is an algebraic sublattice of the lattice \(F(L)\) of all lattice filters of \(L\). The authors endow \(F_n(L)\) with an operation \(\rightsquigarrow\) of relative pseudocomplementation, so that \((F_n(L),\vee,\cap,\rightsquigarrow,\{1\},L)\) is a Heyting algebra. Let \(F^*=F\rightsquigarrow\{1\}\) denote the pseudocomplement of \(F\) in this Heyting algebra.
The authors prove that \((F_n(L),\vee,\cap,^*,\{1\},L)\) is a Boolean algebra if and only if every \(n\)-filter of \(L\) is principal and for every \(x\in L\) there is \(y\in L\) such that \(x\vee y=1\) and \(\varphi_1(x\wedge y)=0\). Further let \(\text{Spec}_n(L)\) denote the set of meet-irreducible elements of \(F_n(L)\). Several characterizations of the filters in \(\text{Spec}_n(L)\) are provided and it is proved that if every \(F\in F_n(L)\) has a unique representation as an intersection of elements of \(\text{Spec}_n(L)\), then again \((F_n(L),\vee,\cap,^*,\{1\},L)\) is a Boolean algebra. The completely meet-irreducible elements of \(F_n(L)\) are characterized as \(n\)-filters \(F\) maximal with respect to some \(A\in L\backslash F\).

MSC:

06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
03G20 Logical aspects of Łukasiewicz and Post algebras
06D20 Heyting algebras (lattice-theoretic aspects)
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References:

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