Ockham algebra

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

• ${\displaystyle \sim (x\wedge y)={}\sim x\vee {}\sim y}$,
• ${\displaystyle \sim (x\vee y)={}\sim x\wedge {}\sim y}$,
• ${\displaystyle \sim 0=1}$,
• ${\displaystyle \sim 1=0}$.

They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety.

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

References

• Berman, Joel (1977), "Distributive lattices with an additional unary operation", Aequationes Mathematicae, 16 (1): 165–171, doi:10.1007/BF01837887, ISSN 0001-9054, MR 0480238 (pdf available from GDZ)
• Blyth, Thomas Scott (2001) [1994], "Ockham algebra", Encyclopedia of Mathematics, EMS Press
• Blyth, Thomas Scott; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. ISBN 978-0-19-859938-8.
• Urquhart, Alasdair (1979), "Distributive lattices with a dual homomorphic operation", Polska Akademia Nauk. Institut Filozofii i Socijologii. Studia Logica, 38 (2): 201–209, doi:10.1007/BF00370442, hdl:10338.dmlcz/102014, ISSN 0039-3215, MR 0544616