The Involution on the Closure Operators

Introduction

A closure operator on a poset is a monotone, extensive, idempotent map; an interior operator is monotone, deflative and idempotent, and the two are dual to each other in the order. When the poset carries an order-reversing involution ${}^{\perp}$, the operator-layer involution of Involutions of the Operator Layer sends a closure operator to the map $c^{\perp} = {}^{\perp} \circ c \circ {}^{\perp}$, and this article shows that $c^{\perp}$ is an interior operator, that the assignment exchanges the two kinds of operator, and that it is the involution that reverses the Galois connection from which the closure operator comes: if $c = g \circ f$ for a Galois connection $f \dashv g$, then $c^{\perp}$ is the interior operator of the reversed connection $g^{\perp} \dashv f^{\perp}$. The fixed closure operators are those commuting with the involution, and on a nondegenerate lattice the only fixed closure operator is the identity. This closes the - * Operator Theory group and returns, with the involution acting on the operators, to the closure operators of Closure Operators and the Consequence Operator.

The article presupposes Closure Operators and the Consequence Operator, for the closure and the Galois connection, Operators on a Poset, for the residuation, Orthocomplemented Lattices and the Involution, for the order-reversing involution, and Involutions of the Operator Layer, for the conjugation. The topological closure and the interior of a topological space are Part II and are named only; nothing topological is used.

The Conjugate of a Closure Operator

Definition. Let $L$ be a poset with an order-reversing involution ${}^{\perp}$. For a closure operator $c$ on $L$, its conjugate is $c^{\perp} = {}^{\perp} \circ c \circ {}^{\perp}$.

Theorem. If $c$ is a closure operator, then $c^{\perp}$ is an interior operator; if $i$ is an interior operator, then $i^{\perp}$ is a closure operator. The assignment $c \mapsto c^{\perp}$ is an involution that exchanges the closure operators and the interior operators of $L$.

Proof. Let $c$ be monotone, extensive ($c(x) \geq x$) and idempotent. The map $c^{\perp}$ is monotone by Involutions of the Operator Layer (a conjugation by an order-reversing involution preserves monotonicity). It is idempotent, $(c^{\perp})^2 = {}^{\perp} c^2 {}^{\perp} = {}^{\perp} c {}^{\perp} = c^{\perp}$. For the deflation, $c(x^{\perp}) \geq x^{\perp}$, and applying the order-reversing ${}^{\perp}$ gives $c^{\perp}(x) = (c(x^{\perp}))^{\perp} \leq (x^{\perp})^{\perp} = x$. By duality, if $c$ is a closure operator with $c \geq \mathrm{id}$ and $c^2 = c$, then $c^{\perp}$ is monotone, idempotent and $\leq \mathrm{id}$, hence an interior operator. The same computation with $c$ replaced by an interior operator $i$ shows that $i^{\perp}$ is a closure operator; and the assignment is an involution because ${}^{\perp}$ is.

Corollary (the open and closed sets). The open sets of $c^{\perp}$, the fixed points of the interior operator, are exactly the complements of the closed sets of $c$: $c^{\perp}(y) = y$ if and only if $y = x^{\perp}$ for a fixed point $x$ of $c$. The fixed points of $c$ and of $c^{\perp}$ correspond under ${}^{\perp}$.

Proof. $c^{\perp}(y) = y$ means $(c(y^{\perp}))^{\perp} = y$, that is, $c(y^{\perp}) = y^{\perp}$; writing $x = y^{\perp}$, the fixed points $y$ of $c^{\perp}$ are exactly the $x^{\perp}$ with $x$ fixed by $c$.

The Reversed Galois Connection

Theorem. Let $f \dashv g$ be a Galois connection between posets with order-reversing involutions, and let $c = g \circ f$ be its closure operator. Then the conjugate $c^{\perp}$ is the interior operator of the reversed connection $g^{\perp} \dashv f^{\perp}$:

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

Proof. The conjugation is an anti-automorphism of composition, $(g \circ f)^{\perp} = f^{\perp} \circ g^{\perp}$, and by Involutions of the Operator Layer the conjugation reverses the adjunction, sending $f \dashv g$ to $g^{\perp} \dashv f^{\perp}$. So $c^{\perp} = f^{\perp} \circ g^{\perp}$ is the composite of a left adjoint followed by a right adjoint — the right adjoint of the reversed connection after its left adjoint — which is the interior operator $f^{\perp} \circ g^{\perp}$ of the reversed connection.

Corollary. The involution reverses the Galois connection of $c$: the left adjoint of the reversed connection is $g^{\perp}$, the conjugate of the right adjoint of the original, and the right adjoint of the reversed connection is $f^{\perp}$, the conjugate of the original left adjoint. Hence the involution on the closure operators is induced by the involution on the Galois connections, and the passage from the closure to the interior is the passage from a connection to the reversed connection.

Proof. The theorem gives the adjunction $g^{\perp} \dashv f^{\perp}$; reading its two members identifies the left and right adjoints as displayed.

The Fixed Closure Operators

Theorem. A closure operator $c$ is fixed by the involution, $c^{\perp} = c$, if and only if $c$ commutes with ${}^{\perp}$. On a nondegenerate lattice with an orthocomplement the only fixed closure operator is the identity, and the only fixed interior operator is the identity.

Proof. $c^{\perp} = {}^{\perp} c {}^{\perp} = c$ is exactly $c \circ {}^{\perp} = {}^{\perp} \circ c$, by composing with ${}^{\perp}$ and using that it is an involution. If $c = c^{\perp}$, then $c$ is at the same time extensive (as a closure operator) and deflative (as its own conjugate, an interior operator), so $c = \mathrm{id}$; the same argument applies to an interior operator.

Corollary. The involution of the closure operators has a trivial fixed part on a nondegenerate lattice, and its orbits are the pairs $\{c, c^{\perp}\}$ of a closure operator and the dual interior operator, together with the identity, which is fixed.

Proof. The fixed part is the identity by the theorem; every other orbit is the pair of an operator and its conjugate, which are distinct because a map that is both extensive and deflative is the identity.

Example (a poset). Let $L$ be the power set of a poset $P$ ordered by inclusion, with the complement of Involutive Set Theory and the Symmetric Difference, and let $c$ be the down-closure, assigning to $A$ the set of elements below a member of $A$. Then $c$ is a closure operator and $c^{\perp}$ is the map assigning to $A$ the complement of the down-closure of the complement, which is the up-closure, an interior operator; the closed sets of $c$ are the down-sets and the open sets of $c^{\perp}$ are the up-sets. The two families correspond under the complement.

Summary

On a poset with an order-reversing involution the conjugation $c \mapsto c^{\perp} = {}^{\perp} \circ c \circ {}^{\perp}$ sends a closure operator to an interior operator and back, and it is an involution exchanging the two families. The open sets of $c^{\perp}$ are the complements of the closed sets of $c$, and if $c = g \circ f$ comes from a Galois connection $f \dashv g$, then $c^{\perp} = f^{\perp} \circ g^{\perp}$ is the interior operator of the **reversed** connection $g^{\perp} \dashv f^{\perp}$; so the involution reverses the Galois connection and exchanges the left and right adjoints. A closure operator is fixed exactly when it commutes with the involution, and on a nondegenerate lattice the only fixed closure operator is the identity, so the involution pairs each closure operator with its dual interior operator.

Summary of Notation

Symbol	Meaning
$c$	A closure operator: monotone, extensive, idempotent
$i$	An interior operator: monotone, deflative, idempotent
$c^{\perp} = {}^{\perp} \circ c \circ {}^{\perp}$	The conjugate of $c$, an interior operator
$f \dashv g$	A Galois connection with closure operator $c = g \circ f$
$g^{\perp} \dashv f^{\perp}$	The reversed connection, with interior operator $c^{\perp}$
fixed part of the involution	Closure operators commuting with ${}^{\perp}$; only the identity