Monadic Orthoframes: Completions and Duality

• Monadic Orthoframes are relational structures pairing an orthogonality relation with a monadic operator, key for representing monadic ortholattices.
• They enable both MacNeille and canonical completions by generating bi-orthogonally closed sets that reconstruct original algebraic structures.
• Generalizations extend these frames to quantum monadic algebras, providing a relational (Kripke-style) duality between algebraic and topological formulations.

Searching arXiv for the cited monadic orthoframe papers and closely related work. arXiv_search(query="monadic ortholattices monadic orthoframes completions duality", max_results=5) Monadic orthoframes are relational structures used in the study of monadic ortholattices, their completions, and their duality theory. In the 2024 treatment of monadic ortholattices, a monadic orthoframe is introduced as a set with an orthogonality relation and an additional binary relation satisfying certain conditions; the MacNeille completion and the canonical completion of a monadic ortholattice are both obtained by forming an associated dual space of this kind, and a suitable topology yields a dual adjunction between monadic ortholattices and monadic orthospaces (Harding et al., 2024). Subsequent work generalizes the MacLaren and Goldblatt constructions to bounded quasi-implication algebras with an additional operator, producing monadic orthoframes that reconstruct the original monadic ortholattice induced by the algebra and furnish relational representations of quantum monadic algebras (McDonald, 23 Jul 2025).

1. Definition and axiomatic content

Let $X$ be a set, let $\perp\subseteq X\times X$ be an irreflexive, symmetric relation, and let $E\subseteq X\times X$ be a binary relation. Then $(X,\perp,E)$ is a monadic orthoframe when the following hold (Harding et al., 2024):

• Orthogonality axioms:

$\forall x\in X\colon\ \neg(x\perp x)$, and $\forall x,y\in X\colon\ x\perp y\iff y\perp x$.

• “Monadic” relation axioms:

$\forall x\in X\colon\ x\,E\,x$, $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$, and

$$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$

For $A\subseteq X$, the associated operators are

$\perp\subseteq X\times X$0

The compatibility condition

$\perp\subseteq X\times X$1

is the axiom that links the “monadic” relation to orthogonality.

The 2025 note uses the notation $\perp\subseteq X\times X$2 and states the same pattern: $\perp\subseteq X\times X$3 is an orthoframe, $\perp\subseteq X\times X$4 is reflexive and transitive, and for each $\perp\subseteq X\times X$5, $\perp\subseteq X\times X$6 (McDonald, 23 Jul 2025). This expresses the same structural idea with $\perp\subseteq X\times X$7 in place of $\perp\subseteq X\times X$8.

2. Monadic orthoframes arising from completions

A monadic ortholattice is a bounded lattice with an order-reversing involution $\perp\subseteq X\times X$9 and a closure-operator $E\subseteq X\times X$0 whose fixed points form a sub-ortholattice. Two associated monadic orthoframes are singled out in the completion theory (Harding et al., 2024).

Construction	Underlying set	Relations
MacNeille completion / MacLaren frame	$E\subseteq X\times X$1	$E\subseteq X\times X$2, $E\subseteq X\times X$3
Canonical completion / Goldblatt frame	$E\subseteq X\times X$4, the proper filters of $E\subseteq X\times X$5	$E\subseteq X\times X$6, $E\subseteq X\times X$7, $E\subseteq X\times X$8

For the MacNeille completion, the bi-orthogonally closed subsets

$E\subseteq X\times X$9

form a complete ortholattice under inclusion, with complement $(X,\perp,E)$0. The quantifier extends by

$(X,\perp,E)$1

The embedding

$(X,\perp,E)$2

is the MacNeille completion of $(X,\perp,E)$3 and preserves $(X,\perp,E)$4 and $(X,\perp,E)$5 (Harding et al., 2024).

For the canonical completion, $(X,\perp,E)$6 is endowed with the Stone topology having sub-basis

$(X,\perp,E)$7

Then $(X,\perp,E)$8 is a monadic orthoframe, called the Goldblatt frame. Its bi-orthogonally closed, clopen subsets

$(X,\perp,E)$9

form the canonical extension $\forall x\in X\colon\ \neg(x\perp x)$0, and the embedding

$\forall x\in X\colon\ \neg(x\perp x)$1

preserves meets, joins, complement, and quantifier, and exhibits the compactness/density conditions of the canonical extension (Harding et al., 2024).

3. Bi-orthogonal closure and completion theorems

The basic algebra associated with a monadic orthoframe is the ortholattice of bi-orthogonally closed subsets. In the MacNeille setting, this is $\forall x\in X\colon\ \neg(x\perp x)$2; in the canonical setting, it is the clopen fragment $\forall x\in X\colon\ \neg(x\perp x)$3. In both cases the involution is

$\forall x\in X\colon\ \neg(x\perp x)$4

and the monadic operation is

$\forall x\in X\colon\ \neg(x\perp x)$5

Theorem 4.1 states that if $\forall x\in X\colon\ \neg(x\perp x)$6 is a monadic ortholattice and $\forall x\in X\colon\ \neg(x\perp x)$7 is its MacNeille embedding, then $\forall x\in X\colon\ \neg(x\perp x)$8, with the operations

$\forall x\in X\colon\ \neg(x\perp x)$9

is the MacNeille completion of $\forall x,y\in X\colon\ x\perp y\iff y\perp x$0 in the variety of monadic ortholattices (Harding et al., 2024). The proof sketch proceeds by showing that every bi-orthogonally closed $\forall x,y\in X\colon\ x\perp y\iff y\perp x$1 is both a join of principal $\forall x,y\in X\colon\ x\perp y\iff y\perp x$2 and a meet of such, while monotonicity of $\forall x,y\in X\colon\ x\perp y\iff y\perp x$3 and the compatibility axiom ensure that $\forall x,y\in X\colon\ x\perp y\iff y\perp x$4 extends to a closure operator on $\forall x,y\in X\colon\ x\perp y\iff y\perp x$5.

Theorem 4.2 states that if $\forall x,y\in X\colon\ x\perp y\iff y\perp x$6 is a monadic ortholattice and $\forall x,y\in X\colon\ x\perp y\iff y\perp x$7 is its filter embedding, then

$\forall x,y\in X\colon\ x\perp y\iff y\perp x$8

is the canonical extension $\forall x,y\in X\colon\ x\perp y\iff y\perp x$9 (Harding et al., 2024). The proof sketch checks density, namely that every $\forall x\in X\colon\ x\,E\,x$0 is a join of meets of $\forall x\in X\colon\ x\,E\,x$1, and compactness, namely finite interpolation of inequalities, in the Stone topology. The formula $\forall x\in X\colon\ x\,E\,x$2 agrees with the standard $\forall x\in X\colon\ x\,E\,x$3-extension of $\forall x\in X\colon\ x\,E\,x$4.

The corollary is that the variety of monadic ortholattices is closed under both MacNeille and canonical completions. Hence every monadic ortholattice embeds into a complete, atomic one arising as bi-orthogonally closed sets of a suitable monadic orthoframe (Harding et al., 2024). This identifies monadic orthoframes as the relational carriers through which both completion procedures are realized.

4. Topology, monadic orthospaces, and dual adjunction

With the introduction of a suitable topology on an orthoframe, a monadic orthoframe becomes part of a monadic orthospace. The relevant category $\forall x\in X\colon\ x\,E\,x$5 has objects $\forall x\in X\colon\ x\,E\,x$6, where $\forall x\in X\colon\ x\,E\,x$7 is the specialization order of $\forall x\in X\colon\ x\,E\,x$8, and morphisms are continuous maps $\forall x\in X\colon\ x\,E\,x$9 that preserve $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$0 and satisfy

$$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$1

on the clopen orthoclosed sets. The algebraic category $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$2 consists of monadic ortholattices and homomorphisms preserving $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$3 (Harding et al., 2024).

The contravariant functors are

$$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$4

and

$$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$5

For each monadic ortholattice $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$6 and monadic orthospace $$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$7 there are mutually inverse bijections

$$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$8

given by

$$\forall x,y,z\in X\colon\ (x\,E\,y\wedge y\,E\,z)\implies x\,E\,z$$9

The unit at $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$0 is the canonical map $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$1, and the counit at $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$2 is $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$3. Hence

$$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$4

is a contravariant adjunction (Harding et al., 2024).

A restriction of this dual adjunction provides a dual equivalence: if $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$5 is restricted to the full subcategory of ortho-sober monadic orthospaces, namely those $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$6 for which every proper filter of $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$7 is $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$8 for a unique $$\forall x\in X\colon\quad E\bigl[E[\{x\}]^{\perp}\bigr]\subseteq E[\{x\}]^{\perp}.$$9, then $A\subseteq X$0 and $A\subseteq X$1 are isomorphisms, and $A\subseteq X$2 and $A\subseteq X$3 furnish a dual equivalence (Harding et al., 2024).

5. Examples and comparative remarks

A basic example is obtained by taking $A\subseteq X$4 with $A\subseteq X$5, reflexive $A\subseteq X$6 given by all pairs, and the discrete topology. Then $A\subseteq X$7 is a monadic orthoframe, and its closed sets under bi-orthogonality are

$A\subseteq X$8

which form a 4-element Boolean monadic ortholattice (Harding et al., 2024).

This example displays the elementary mechanism of the representation theory: the relational data $A\subseteq X$9 determine bi-orthogonally closed subsets, and those subsets inherit both an ortholattice complement $\perp\subseteq X\times X$00 and a monadic operator $\perp\subseteq X\times X$01. In the two-point case, the resulting algebra is Boolean; in the general case, the same construction applies to non-Boolean ortholattices.

The comparison explicitly noted in the 2024 treatment is with Halmos’ Stone–Montague duality for Boolean monadic algebras, replacing ultrafilters by proper filters and Boolean posets by ortholattices with involution (Harding et al., 2024). This indicates that monadic orthoframes play, for monadic ortholattices, the role that relational Stone-style spaces play for Boolean monadic structures.

6. Generalization to monadic quasi-implication algebras

The 2025 note extends the monadic orthoframe constructions to bounded quasi-implication algebras with an additional operator, called monadic quasi-implication algebras. A quantum monadic algebra is an orthomodular lattice equipped with a closure operator, known as a quantifier, whose closed elements form an orthomodular sub-lattice. Every quantum monadic algebra can be converted into a monadic quasi-implication algebra with the underlying magma structure determined by the operation of Sasaki implication on the underlying orthomodular lattice, and every monadic quasi-implication algebra can be converted into a quantum monadic algebra; these constructions induce an isomorphism between the category of quantum monadic algebras and the category of monadic quasi-implication algebras (McDonald, 23 Jul 2025).

Within this setting, two monadic orthoframes are defined. The monadic MacLaren frame of $\perp\subseteq X\times X$02 is

$\perp\subseteq X\times X$03

where

$\perp\subseteq X\times X$04

Theorem (5.9) states that $\perp\subseteq X\times X$05 is a monadic orthoframe (McDonald, 23 Jul 2025).

The monadic Goldblatt frame of $\perp\subseteq X\times X$06 is

$\perp\subseteq X\times X$07

where $\perp\subseteq X\times X$08 are the proper filters of $\perp\subseteq X\times X$09,

$\perp\subseteq X\times X$10

Theorem (5.13) states that $\perp\subseteq X\times X$11 is a monadic orthoframe (McDonald, 23 Jul 2025).

In each case, the complete lattice $\perp\subseteq X\times X$12 of bi-orthogonally closed sets, equipped with

$\perp\subseteq X\times X$13

reconstructs the original monadic ortholattice induced by $\perp\subseteq X\times X$14. Hence these frames furnish relational (Kripke-style) representations of monadic quasi-implication algebras and quantum monadic algebras (McDonald, 23 Jul 2025). A plausible implication is that the monadic orthoframe formalism is not confined to one algebraic presentation of quantum monadic structure, but is stable under passage between equivalent categorical formulations.