Residuated mapping

In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function.

If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ↓{b} = { b' ∈ B : b' ≤ b }. In general the preimage under f of a principal down-set need not be a principal down-set. If it is, f is called residuated.

The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One speaks only of residuated algebra for higher arities). A binary (or higher arity) residuated map is usually not residuated as a unary map.^[1]

Definition

If A, B are posets, a function f: A → B is residuated if and only if the preimage under f of every principal down-set of B is a principal down-set of A.

Consequences

With A, B posets, the set of functions A → B can be ordered by the pointwise order f ≤ g ↔ (∀x ∈ A) f(x) ≤ g(x).

It can be shown that f is residuated if and only if there exists a (necessarily unique) monotone function f ⁺: B → A such that f o f ⁺ ≤ id_B and f ⁺ o f ≥ id_A, where id is the identity function. The function f ⁺ is the residual of f. A residuated function and its residual form a Galois connection under the (more recent) monotone definition of that concept, and for every (monotone) Galois connection the lower adjoints is residuated with the residual being the upper adjoint.^[2] Therefore, the notions of monotone Galois connection and residuated mapping essentially coincide.

Additionally, we have f ^-1(↓{b}) = ↓{f ⁺(b)}.

If B° denotes the order dual (opposite poset) to B then f : A → B is a residuated mapping if and only if f : A → B° and f ^*: B° → A form a Galois connection under the original antitone definition of this notion.

If f : A → B and g : B → C are residuated mappings, then so is the function composition fg : A → C, with residual (fg) ⁺ = g ⁺f ⁺. The antitone Galois connections do not share this property.

The set of monotone transformations (functions) over a poset is an ordered monoid with the pointwise order, and so is the set of residuated transformations.^[3]

Examples

• The ceiling function $x \mapsto \lceil x \rceil$ from R to Z (with the usual order in each case) is residuated, with residual mapping the natural embedding of Z into R.
• The embedding of Z into R is also residuated. Its residual is the floor function $x \mapsto \lfloor x \rfloor$.

Residuated binary operators

If • : P × Q → R is a binary map and P, Q, and R are posets, then one may define residuation component-wise for the left and right translations, i.e. multiplication by a fixed element. For an element x in P define _xλ(y) = x • y, and for x in Q define λ_x(y) = y • x. Then • is said to be residuated if and only if _xλ and λ_x are residuated for all x (in P and respectively Q). Left (and resp. right) division are defined by taking the residuals of the left (and resp. right) translations: x\y = (_xλ)⁺(y) and x/y = (λ_x)⁺(y)

For example, every ordered group is residuated, and the division defined by the above coincides with notion of division in a group. A less trivial example is the set Mat_n(B) of square matrices over a boolean algebra B, where the matrices are ordered pointwise. The pointwise order endows Mat_n(B) with pointwise meets, joins and complements. Matrix multiplication is defined in the usual manner with the "product" being a meet, and the "sum" a join. It can be shown^[4] that X\Y = (Y^tX’)’ and X/Y = (X’Y^t)’, where (X’ is the complement of X, and Y^t is the transposed matrix).

See also

• Residuated lattice

Notes

1. Denecke, p. 95; Galatos, p. 148
2. Erné, Proposition 4
3. Blyth, 2005, p. 193
4. Blyth, p. 198

References

• J.C. Derderian, "Galois connections and pair algebras", Canadian J. Math. 21 (1969) 498-501.
• Jonathan S. Golan, Semirings and Affine Equations Over Them: Theory and Applications, Kluwer Academic, 2003, ISBN 1-4020-1358-2. Page 49.
• T.S. Blyth, "Residuated mappings", Order 1 (1984) 187-204.
• T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5. Page 7.
• T.S. Blyth, M. F. Janowitz, Residuation Theory, Pergamon Press, 1972, ISBN 0-08-016408-0. Page 9.
• M. Erné, J. Koslowski, A. Melton, G. E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103–125. Available online in various file formats: PS.GZ PS
• Klaus Denecke, Marcel Erné, Shelly L. Wismath, Galois connections and applications, Springer, 2004, ISBN 1402018975
• Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono (2007), Residuated Lattices. An Algebraic Glimpse at Substructural Logics, Elsevier, ISBN 978-0-444-52141-5.