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Classification of weak De Morgan algebras. (English) Zbl 0835.06012

Summary: We first show that for every weak De Morgan algebra \(L(n)\) of order \(n\) (WDM-\(n\) algebra), there is a quotient weak De Morgan algebra \(L(n) /\sim\) which is embeddable in the finite WDM-\(n\) algebra \(\Omega (n)\). We then demonstrate that the finite WDM-\(n\) algebra \(\Omega (n)\) is functionally free for the class \(\text{CL} (n)\) of WDM-\(n\) algebras. That is, we show that any formulas \(f\) and \(g\) are identically equal in each algebra in \(\text{CL} (n)\) if and only if they are identically equal in \(\Omega (n)\). Finally we establish that there is no weak De Morgan algebra whose quotient algebra by a maximal filter has exactly seven elements.

MSC:

06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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References:

[1] Kondo, M., “Representation theorem of quasi-Kleene algebras in terms of Kripke-type frames,” Mathematica Japonica , vol. 38 (1993), pp. 185–189. · Zbl 0771.06004
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