Boolean algebra

:For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic.

Order theoretic properties

Like any lattice, a Boolean algebra (A, land, lor) gives rise to a partially ordered set (A, ≤) by defining

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: a ≤ b iff a = a land b

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(which is also equivalent to b = a lor b).

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In fact one can also define a Boolean algebra to be a distributive lattice (A, ≤) (considered as a partially ordered set) with least element 0 and greatest element 1, within which every element x has a complement ¬x such that

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: x land ¬x = 0 and x lor ¬x = 1

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Here land and lor are used to denote the infimum (meet) and supremum (join) of two elements. Again, if complements in the above sense exist, then they are uniquely determined.

Related Topics:
Infimum - Supremum

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The algebraic and the order theoretic perspective can usually can be used interchangeably and both are of great use to import results and concepts from both universal algebra and order theory. In many practical examples an ordering relation, conjunction, disjunction, and negation are all naturally available, so that it is straightforward to exploit this relationship.

Related Topics:
Universal algebra - Order theory

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