Ockham Congruences Whose Quotient Algebras Are Boolean

Abstract

Given any Ockham algebra, we describe the congruences such that the quotient algebras are boolean. This description is obtained using certain ideals that we call pro-boolean ideals. We prove that every proper pro-boolean ideal is the intersection of a family of falsity ideals. We also determine when every proper pro-boolean ideal is a unique intersection of such ideals. Finally, we show that if an Ockham algebra in the Urquhart class P _n+2,n is fixed point free then the corresponding dual space has a fixed point. This result is a natural generalisation of a well known theorem (Blyth, T. S., Varlet, J. C. (1994). Ockham Algebras. Oxford University Press, Theorem 6.3).

Key Words:

• Ockham algebra
• Congruence
• Falsity ideal
• pro-boolean ideal

1991 Mathematics Subject Classification:

• 06D30

Acknowledgment

We are grateful to Professor Blyth for helpful discussions on the presentation of the results in this paper.