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residuated lattice

Definition

A residuated lattice is an algebra, L=(L,,,,\,/,1)L=(L,\wedge,\vee, \cdot, \backslash, /,1) such that

  • (L,,)(L,\wedge,\vee) is a lattice;

  • (L,,1)(L,\cdot,1) is a monoid;

  • for all a,b,cLa,b,c\in L,

(abc)(ba\c)(ac/b).(ab\le c)\Leftrightarrow (b\le a \backslash c)\Leftrightarrow (a\le c / b).

Examples

Powerset of a monoid.

Let (M,,e)(M,\cdot, e) be a monoid and consider L=𝒫(M)L=\mathcal{P}(M).

If X,YMX, Y\subseteq M, define

XY={xyxX,yY};X\cdot Y=\{x\cdot y\mid x\in X, y\in Y\};
X\Y={zMX{z}Y};X\backslash Y = \{z\in M\mid X\cdot\{z\}\subseteq Y\};
Y/X={zM{z}XY}Y/ X= \{z\in M\mid \{z\}\cdot X\subseteq Y\}

and then LL with this structure is a residuated lattice.

  • A special instance of this is with the quantale of ideals in a ring, RR; see ideals in a monoid and quantale, especially the section on examples.

Relation algebra on a set

For any set, XX, the relation algebra, Rel(X)=𝒫(X×X)Rel(X)=\mathcal{P}(X\times X) of binary relations on XX is a residuated lattice.

Writing R R^- for the complement of a relation, RR, and R;SR;S for the relational composition of two relations, RR and SS, we define

R\S=(R;S ) ,R \backslash S = (R;S^-)^- ,

S/R=(S ;R) S / R = (S^-;R)^-

and

RS=(RS ) =R S.R\to S= (R\cap S^-)^- = R^-\cup S.

We have

  • (Rel(X),,,,,X 2)(Rel(X), \cap,\cup,\to, \emptyset, X^2) is a Boolean algebra,

  • (Rel(X),;Δ)(Rel(X), ; \Delta) is a monad, where Δ\Delta is the diagonal / equality relation;

  • the residuation relation is satisfied:

((R;S)T)(S(R\T))(RT/S)).((R;S)\subseteq T)\Leftrightarrow (S\subseteq (R\backslash T))\Leftrightarrow (R\subseteq T / S)).

Lattice-ordered group

Any lattice ordered group gives a residuated lattice. This is described in the entry on lattice ordered groups. It is noteworthy that in this context, the residuals are quotients and, in fact, the inverses in the group are residuals.

Categorical interpretations of residuation

Looking at the poset structure of LL as a category, one rewrites aba\le b as aba\to b and L(a,b)L(a,b) for the singleton set consisting just of the morphism aba\to b.

We have for each element aa in LL, two mappings given by, respectively, left and right multiplication by aa:

λ a:xax\lambda_a:x\mapsto a\cdot x
ρ a:xxa.\rho_a:x\mapsto x\cdot a.

These mappings are order preserving, so give endofunctors on the category LL. For instance, suppose xyx\le y and we have ayba\cdot y\leq b. We then have ya/by\leq a/b, and hence xa/bx\le a/b, i.e. axba\cdot x\leq b. Now take b=ayb= a\cdot y. We have thus that axaya\cdot x\leq a\cdot y, as required.

In this interpretation, both these functors have right adjoints, namely the two residuals. We have, for instance,

L(λ a(x),c)L(x,a\c).L(\lambda_a(x),c) \cong L(x,a\backslash c).

The residual functor, a\a\backslash -, acts like a left exponential object for the left multiplication functor. Likewise /a-/a acts like a right exponential object functor.

This point of view is explored further in the entry on residuals.

References

A classical reference is

  • M. Ward and R. P. Dilworth, Residuated Lattices, Transactions of the American Mathematical Society, Vol. 45, No. 3 (May, 1939), pp. 335-354.doi.org/10.2307/1990008

A set of slides from a modern treatment is

  • Nikolaos Galatos, Residuated lattices, slides

Last revised on September 10, 2021 at 03:19:50. See the history of this page for a list of all contributions to it.