Editing Boolean algebra (structure) (section)

== Boolean rings ==
{{Main|Boolean ring}}
Every Boolean algebra {{math|1=(''A'', ∧, ∨)}} gives rise to a [[ring (algebra)|ring]] {{math|(''A'', +, ·)}} by defining {{math|1=''a'' + ''b'' := (''a'' ∧ ¬''b'') ∨ (''b'' ∧ ¬''a'') = (''a'' ∨ ''b'') ∧ ¬(''a'' ∧ ''b'')}} (this operation is called [[symmetric difference]] in the case of sets and [[Truth table#Exclusive disjunction|XOR]] in the case of logic) and {{math|1=''a'' · ''b'' := ''a'' ∧ ''b''}}. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the {{math|1}} of the Boolean algebra. This ring has the property that {{math|1=''a'' · ''a'' = ''a''}} for all {{math|''a''}} in {{math|''A''}}; rings with this property are called [[Boolean ring]]s.

Conversely, if a Boolean ring {{math|''A''}} is given, we can turn it into a Boolean algebra by defining {{math|1=''x'' ∨ ''y'' := ''x'' + ''y'' + (''x'' · ''y'')}} and {{math|1=''x'' ∧ ''y'' := ''x'' · ''y''}}.{{sfn|Stone|1936}}{{sfn|Hsiang|1985|p=260}}
Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map {{math|''f'' : ''A'' → ''B''}} is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The [[category theory|categories]] of Boolean rings and Boolean algebras are [[equivalence of categories|equivalent]];{{sfn|Cohn|2003|p=[https://books.google.com/books?id=VESm0MJOiDQC&pg=PA81 81]}} in fact the categories are [[Isomorphism of categories|isomorphic]].

Hsiang (1985) gave a [[Abstract rewriting system|rule-based algorithm]] to [[Word problem (mathematics)|check]] whether two arbitrary expressions denote the same value in every Boolean ring.
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Hsiang (1985) gave a [[Confluence (abstract rewriting)|confluent]] and [[Abstract rewriting system#Termination and convergence|terminating]] [[Abstract rewriting system|rewrite system]] for Boolean rings, thus solving their [[Word problem (mathematics)|word problem]]: to check whether two arbitrary expressions ''s'' and ''t'' denote the same value in every Boolean ring, apply rewrite rules to ''s'' as long as possible, resulting in an expression ''s''<sub>n</sub>, obtain ''t''<sub>n</sub> from ''t'' in a similar way, and check whether ''s''<sub>n</sub> and ''t''<sub>n</sub> are literally identical, except for different parenthezation and order of operands of "+" or "·".
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More generally, Boudet, [[Jean-Pierre Jouannaud|Jouannaud]], and Schmidt-Schauß (1989) gave an algorithm to [[Unification (computer science)#Particular background knowledge sets E|solve equations]] between arbitrary Boolean-ring expressions.
Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in [[automated theorem proving]].