Ockham algebra
Encyclopedia
In mathematics, an Ockham algebra is a bounded distributive lattice
with a dual endomorphism
. They were introduced by , and were named after William of Ockham
by .
Examples of Ockham algebras include Boolean algebra
s, De Morgan algebra
s, Stone algebra
s, and Kleene algebra
s.
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
with a dual endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
. They were introduced by , and were named after William of Ockham
William of Ockham
William of Ockham was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of...
by .
Examples of Ockham algebras include Boolean algebra
Boolean algebra
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...
s, De Morgan algebra
De Morgan algebra
In mathematics, a De Morgan algebra is a structure A = such that:* is a bounded distributive lattice, and...
s, Stone algebra
Stone algebra
In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattices such that a*∪a** = 1. They were introduced by and named after Marshall Harvey Stone....
s, and Kleene algebra
Kleene algebra
In mathematics, a Kleene algebra is either of two different things:* A bounded distributive lattice with an involution satisfying De Morgan's laws , additionally satisfying the inequality x∧−x ≤ y∨−y. Kleene algebras are subclasses of Ockham algebras...
s.