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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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