In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.
Definition
A residuated Boolean algebra is an algebraic structure such that
An equivalent signature better suited to the relation algebra application is where the unary operations x\ and x▷ are intertranslatable in the manner of De Morgan's laws via x\y = ¬(x▷¬y), x▷y = ¬(x\¬y),
and dually /y and ◁y as
x/y = ¬(¬x◁y), x◁y = ¬(¬x/y),
with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z by ¬z) to read ⇔ ⇔
This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy.
Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety.
Examples
Any Boolean algebra, with the monoid multiplication • taken to be conjunction and both residuals taken to be material implication x→y. Of the remaining 15 binary Boolean operations that might be considered in place of conjunction for the monoid multiplication, only five meet the monotonicity requirement, namely 0, 1, x, y, and . Setting y = z = 0 in the residuation axiom y ≤ x\z ⇔ x•y ≤ z, we have 0 ≤ x\0 ⇔ x•0 ≤ 0, which is falsified by taking x = 1 when x•y = 1, x, or . The dual argument for z/y rules out x•y = y. This just leaves x•y = 0 (a constant binary operation independent of x and y), which satisfies almost all the axioms when the residuals are both taken to be the constant operation x/y = x\y = 1. The axiom it fails is , for want of a suitable value for . Hence conjunction is the only binary Boolean operation making the monoid multiplication that of a residuated Boolean algebra. The power set 2X2 made a Boolean algebra as usual with ∩, ∪ and complement relative to X2, and made a monoid with relational composition. The monoid unit is the identity relation { (x,x)|x ∈ X}. The right residual R\S is defined by x(R\S)y if and only if for all z in X, zRx implies zSy. Dually the left residual S/R is defined by y(S/R)x if and only if for all z in X, xRz implies ySz. The power set 2Σ* made a Boolean algebra as for Example 2, but with language concatenation for the monoid. Here the set Σ is used as an alphabet while Σ* denotes the set of all finite (including empty) words over that alphabet. The concatenation LM of languages L and M consists of all words uv such that u ∈ L and v ∈ M. The monoid unit is the language { ε} consisting of just the empty word ε. The right residual M\L consists of all words w over Σ such that Mw ⊆ L. The left residual L/M is the same with wM in place of Mw.
Conjugacy
The De Morgan duals ▷ and ◁ of residuation arise as follows. Among residuated lattices, Boolean algebras are special by virtue of having a complementation operation ¬. This permits an alternative expression of the three inequalities y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y
in the axiomatization of the two residuals in terms of disjointness, via the equivalence x ≤ y ⇔ x∧¬y = 0. Abbreviating x∧y = 0 to x # y as the expression of their disjointness, and substituting ¬z for z in the axioms, they become with a little Boolean manipulation ¬(x\¬z) # y ⇔ x•y # z ⇔ ¬(¬z/y) # x