Ockham Algebra

In mathematics, an Ockham algebra is a bounded distributive lattice $L$ with a dual endomorphism, that is, an operation $\sim : L \to L$ satisfying

$\sim (x \land y) = \sim x \lor \sim y$ ,
$\sim (x \lor y) = \sim x \land \sim y$ ,
$\sim 0 = 1$ ,
$\sim 1 = 0$ .

They were introduced by Berman,^[1] and were named after William of Ockham by Urquhart.^[2] Ockham algebras form a variety.

Examples

Examples of Ockham algebras include Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras.

References

↑ Berman, Joel (February 1977). "Distributive lattices with an additional unary operation" (in en). Aequationes Mathematicae 15 (1): 118–118. doi:10.1007/BF01837887. ISSN 0001-9054. http://link.springer.com/10.1007/BF01837887.
↑ Urquhart, Alasdair (1979). "Distributive lattices with a dual homomorphic operation" (in en). Studia Logica 38 (2): 201–209. doi:10.1007/BF00370442. ISSN 0039-3215. http://link.springer.com/10.1007/BF00370442.

Hazewinkel, Michiel, ed. (2001), "Ockham algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=o/o110030
Blyth, Thomas Scott; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. ISBN 978-0-19-859938-8.

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[1] Berman, Joel (February 1977). "Distributive lattices with an additional unary operation" (in en). Aequationes Mathematicae 15 (1): 118–118. doi:10.1007/BF01837887. ISSN 0001-9054. http://link.springer.com/10.1007/BF01837887.

[2] Urquhart, Alasdair (1979). "Distributive lattices with a dual homomorphic operation" (in en). Studia Logica 38 (2): 201–209. doi:10.1007/BF00370442. ISSN 0039-3215. http://link.springer.com/10.1007/BF00370442.

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Ockham Algebra

Examples

References

Further reading