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Boolean lattice
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(Definition)
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In this entry, the notions of a Boolean lattice, a Boolean algebra, and a Boolean ring are defined, compared and contrasted.
A Boolean lattice  is a distributive lattice in which for each element  there exists a complement  such that
In other words, a Boolean lattice is the same as a complemented distributive lattice. A morphism between two Boolean lattices is just a lattice homomorphism (so that  and  may not be preserved).
A Boolean algebra is a Boolean lattice such that  and 0 are considered as operators (unary and nullary respectively) on the algebraic system. In other words, a morphism (or a Boolean algebra homomorphism) between two Boolean algebras must preserve  and
 . As a result, the category of Boolean algebras and the category of Boolean lattices are not the same (and the former is a subcategory of the latter).
A Boolean ring is an (associative) unital ring  such that for any  ,  . It is easy to see that
Boolean rings (with identity, but allowing 0=1) are equivalent to Boolean lattices. To view a Boolean ring as a Boolean lattice, define
 and   and  The category of Boolean algebras is naturally equivalent to the category of Boolean rings.
- 1
- G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
- 2
- R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
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Cross-references: naturally equivalent, equivalent, identity, commutative ring, characteristic, easy to see, unital ring, associative, subcategory, category, preserve, Boolean algebra homomorphism, algebraic system, unary, operators, lattice homomorphism, Boolean, morphism, complemented, complement, distributive lattice, Boolean ring
There are 61 references to this entry.
This is version 15 of Boolean lattice, born on 2002-02-24, modified 2008-04-29.
Object id is 2594, canonical name is BooleanLattice.
Accessed 19947 times total.
Classification:
| AMS MSC: | 03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures) | | | 06B20 (Order, lattices, ordered algebraic structures :: Lattices :: Varieties of lattices) | | | 03G05 (Mathematical logic and foundations :: Algebraic logic :: Boolean algebras) | | | 06E05 (Order, lattices, ordered algebraic structures :: Boolean algebras :: Structure theory) |
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Pending Errata and Addenda
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