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A generalization of the Łukasiewicz algebras. (English) Zbl 0995.03047
An Ockham algebra is an algebra \(\langle A;\wedge,\vee, f,0,1\rangle\) where \(\langle A;\wedge,\vee, 0,1\rangle\) is a bounded distributive lattice and \(f\) is a dual endomorphism. This notion generalizes the notion of a De Morgan algebra. Łukasiewicz algebras have a reduct which is a De Morgan algebra. In this paper, the authors introduce the variety \({\mathcal L}^m_n\), \(m\geq 1\), \(n\geq 2\), of \(m\)-generalized Łukasiewicz algebras of order \(n\) in which the De Morgan reduct is replaced by a reduct satisfying \(f^{2m}(x)= x\) for some \(m\geq 1\). Some algebraic properties of the variety \({\mathcal L}^m_n\) are investigated and the subdirectly irreducible algebras are characterized. The variety \({\mathcal L}^m_n\) is semisimple, locally finite and has equationally definable principal congruences.

MSC:
03G20 Logical aspects of Łukasiewicz and Post algebras
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06B20 Varieties of lattices
08B26 Subdirect products and subdirect irreducibility
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