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,^[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

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

Further reading

• 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.