Residuated mapping

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 and only if it is order-preserving: that is, "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"' ε "B" : "b"' ≤ "b" }. In general the preimage of a principal down-set need not be a principal down-set.

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 an unique monotone function "f" +: "B" → "A" such that"f" o "f" + ≤ idB and "f" + o "f" ≥ idA, where id is the identity function. The function "f" + is the residual of "f". 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.

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 set of monotone transformations (functions) over a poset is an ordered monoid with the pointwise order, and so is the set of residuated transformations. [Blyth, 2005, p. 193]

Examples

* The floor function x mapsto lfloor x floor from R to Z (with the usual order in each case) is residuated, with residual mapping the natural embedding of Z into R.

ee also

* Residuated lattice

Notes

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 1402013582. 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 0080164080. Page 9.


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