Involutions of the Operator Layer

Introduction

The two preceding articles of this Part put an involution on the elements of a lattice or a relation algebra. This article puts the involution on the operators themselves. Given a poset $P$ with an order-reversing involution $x \mapsto x^{\perp}$, the conjugate of a monotone map $f : P \to P$ is the map $f^{\perp} = {}^{\perp} \circ f \circ {}^{\perp}$, and the assignment $f \mapsto f^{\perp}$ is an involution of the operator layer. It is an anti-automorphism for composition, $(g \circ f)^{\perp} = f^{\perp} \circ g^{\perp}$, and it is order-reversing for the pointwise order; its fixed elements are the operators commuting with the involution, and it exchanges left adjoints with right adjoints, so it reverses every Galois connection. An order-preserving involution $\sigma$ gives the other case, the conjugation $f \mapsto \sigma \circ f \circ \sigma$, which is an order-preserving automorphism of the operator monoid. This article develops both, the fixed part and the adjoint operation.

The article presupposes Operators on a Poset — monotone maps, residuated maps, Galois connections, the monoid of monotone maps — Orthocomplemented Lattices and the Involution, for the order-reversing involution, and Involutive Categories and the Dagger Functor, for the abstract involution on morphisms. The operator-algebraic and Hilbert-space adjoint is Part II and the - * Operator Theory group, and is named only; here the operators are monotone maps of a poset and the involution acts on them.

Conjugation by the Involution

Definition. Let $P$ be a poset with an order-reversing involution ${}^{\perp}$ and let $f : P \to P$ be monotone. The conjugate of $f$ is

$$ f^{\perp} = {}^{\perp} \circ f \circ {}^{\perp} : P \to P, \qquad f^{\perp}(x) = (f(x^{\perp}))^{\perp}. $$

Theorem. The conjugate of a monotone map is monotone, and the assignment $f \mapsto f^{\perp}$ is an involution of the set of monotone maps. It is an anti-automorphism of the composition monoid and an order-reversing map of the pointwise order:

$$ (f^{\perp})^{\perp} = f, \qquad (g \circ f)^{\perp} = f^{\perp} \circ g^{\perp}, \qquad f \leq g \Rightarrow g^{\perp} \leq f^{\perp} . $$

Proof. For monotonicity, if $x \leq y$ then $y^{\perp} \leq x^{\perp}$, so $f(y^{\perp}) \leq f(x^{\perp})$ and $(f(y^{\perp}))^{\perp} \geq (f(x^{\perp}))^{\perp}$; comparing with the definition gives $f^{\perp}(x) \leq f^{\perp}(y)$. The involution is $(f^{\perp})^{\perp} = {}^{\perp} \circ {}^{\perp} \circ f \circ {}^{\perp} \circ {}^{\perp} = f$ because ${}^{\perp}$ has order two. For the composition, $(g \circ f)^{\perp} = {}^{\perp} \circ g \circ f \circ {}^{\perp} = ({}^{\perp} \circ g \circ {}^{\perp}) \circ ({}^{\perp} \circ f \circ {}^{\perp}) = g^{\perp} \circ f^{\perp}$. For the order, $f \leq g$ pointwise gives $f(x^{\perp}) \leq g(x^{\perp})$, and applying the order-reversing ${}^{\perp}$ reverses the inequality, so $g^{\perp}(x) \leq f^{\perp}(x)$ for all $x$.

Corollary. The involution $f \mapsto f^{\perp}$ is an automorphism of the order-dual of the operator monoid: it is a bijection that reverses the pointwise order and the composition. In particular it maps the identity to the identity, the constant maps to the constant maps, and idempotents to idempotents.

Proof. The identity satisfies $\mathrm{id}^{\perp} = \mathrm{id}$; a constant map $c_a$ with value $a$ satisfies $c_a^{\perp} = c_{a^{\perp}}$, so constants go to constants; and $f^2 = f$ implies $(f^{\perp})^2 = (f^2)^{\perp} = f^{\perp}$ by the composition law.

Definition. The fixed part of the involution of the operator layer is the set of operators with $f^{\perp} = f$.

Theorem. A monotone map $f$ is fixed by the conjugation exactly when it commutes with the involution, $f \circ {}^{\perp} = {}^{\perp} \circ f$. The fixed part contains the identity and is closed under composition; it is the centralizer of the involution in the monoid of monotone maps.

Proof. $f^{\perp} = {}^{\perp} \circ f \circ {}^{\perp} = f$ is exactly $f \circ {}^{\perp} = {}^{\perp} \circ f$ after composing with ${}^{\perp}$ on one side and using that it is an involution. The fixed elements of a monoid endomorphism are always a submonoid, which here is the centralizer of ${}^{\perp}$.

The Adjoint Operation

Theorem. The conjugation reverses adjoint pairs: if $f \dashv g$ is a Galois connection between posets with order-reversing involutions, then $g^{\perp} \dashv f^{\perp}$, that is, the left and the right adjoints are exchanged.

Proof. The Galois condition is $f(x) \leq y \Leftrightarrow x \leq g(y)$. Replacing $x$ by $x^{\perp}$ and $y$ by $y^{\perp}$ and using the involutions, $f(x^{\perp}) \leq y^{\perp} \Leftrightarrow x^{\perp} \leq g(y^{\perp})$, which becomes $y \leq f^{\perp}(x) \Leftrightarrow g^{\perp}(y) \leq x$ after applying ${}^{\perp}$ and rewriting; this is the Galois condition $g^{\perp} \dashv f^{\perp}$.

Corollary. The conjugation carries the set of left adjoints bijectively onto the set of right adjoints and exchanges the two; it is the adjoint operation of the scope, an order-reversing involution between the two families of residuated maps.

Proof. A left adjoint $f$ has a right adjoint $g$, and the theorem gives the right adjoint $g^{\perp}$ of $f^{\perp}$; applying the conjugation twice returns $f$ and $g$, so the map is a bijection exchanging the two families.

Example (relations). Let the poset be the lattice of relations on a set with the converse as the involution, in the sense of The Converse as an Adjoint: for an operator $F$ on relations put $F^{\perp}(R) = (F(R^{-1}))^{-1}$. Then $F \mapsto F^{\perp}$ is the conjugation, the fixed operators are those with $F(R^{-1}) = F(R)^{-1}$, that is, those commuting with the converse, and the conjugation is an anti-automorphism of the composition of operators. The left-composition operator $L_R(S) = R;S$ of The Converse as an Adjoint has the conjugate $L_R^{\perp}(S) = (L_R(S^{-1}))^{-1} = (R;S^{-1})^{-1} = S;R^{-1}$, the right-composition by the converse of $R$; this is computed again in The Converse Relation as an Adjoint.

Example (a Boolean lattice). Let $P$ be a Boolean lattice and let ${}^{\perp}$ be the complement. For a monotone map $f$ the conjugate is $f^{\perp}(x) = \neg f(\neg x)$, the dual of $f$; the fixed maps are those commuting with the complement, and the conjugation exchanges the meet-preserving maps with the join-preserving maps, because it reverses the composition and the order.

The Order-Preserving Case

Definition. Let $P$ have an order-preserving involution $\sigma$. The conjugate of a monotone map $f$ is $f^{\sigma} = \sigma \circ f \circ \sigma$.

Theorem. If $\sigma$ is an order-preserving involution, then $f \mapsto f^{\sigma}$ is an involution of the set of monotone maps, an automorphism of the composition monoid, and an order-preserving map of the pointwise order:

$$ (f^{\sigma})^{\sigma} = f, \qquad (g \circ f)^{\sigma} = g^{\sigma} \circ f^{\sigma}, \qquad f \leq g \Rightarrow f^{\sigma} \leq g^{\sigma} . $$

Proof. The same computations as before, with $\sigma$ order-preserving, so that $f \leq g$ gives $\sigma \circ f \circ \sigma \leq \sigma \circ g \circ \sigma$.

Corollary. An order-preserving involution of the base gives an ordinary automorphism of the operator monoid, and its fixed part is again the centralizer of $\sigma$. The order-reversing case is the one relevant to an orthocomplement, and it is the case of the rest of the group.

Proof. The fixed equation is the same $f \circ \sigma = \sigma \circ f$; the last assertion records that an orthocomplement is order-reversing.

Summary

An order-reversing involution ${}^{\perp}$ of a poset acts on the operators by conjugation, $f^{\perp} = {}^{\perp} \circ f \circ {}^{\perp}$; this is an involution of the operator layer, an anti-automorphism of composition and an order-reversing map of the pointwise order, and its fixed elements are the operators commuting with ${}^{\perp}$. It exchanges left and right adjoints, so it reverses every Galois connection, and it carries left adjoints bijectively onto right adjoints. An order-preserving involution $\sigma$ gives instead the conjugation $f \mapsto \sigma \circ f \circ \sigma$, an order-preserving automorphism of the operator monoid with the same fixed part.

Summary of Notation

Further Reading

• Garrett Birkhoff, Lattice Theory, 3rd ed. (American Mathematical Society, 1967), for monotone maps, residuated maps and Galois connections and their duals.
• Marcel Erné, "Adjunctions and Galois connections: origins, history and development", in Galois Connections and Applications (Kluwer, 2004), for the exchange of adjoints under duality.
• Chris Brink, Wolfram Kahl and Gunther Schmidt, Relational Methods in Computer Science (Springer, 1997), for the conjugation of operators on relations by the converse.
• Brian A. Davey and Hilary A. Priestley, Introduction to Lattices and Order, 2nd ed. (Cambridge University Press, 2002), for monotone maps, adjoints and the duality of Galois connections.
• Richard Bird and Oege de Moor, Algebra of Programming (Prentice Hall, 1997), for the dual of a map under an involution and the algebra of operators.