---
title: Monadic Quasi-Implication Algebras
url: https://www.emergentmind.com/topics/monadic-quasi-implication-algebras
type: topic
---

# Monadic Quasi-Implication Algebras

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Monadic quasi-implication algebras are algebras of type $\langle 2,0,1\rangle$ of the form $\langle A;\cdot,0,\Diamond\rangle$ in which the binary operation $\cdot$ is the quasi-implication operation of a bounded quasi-implication algebra and the unary operation $\Diamond$ is a monadic operator tailored to the non-distributive setting of orthomodular lattices. In the formulation developed in "Orthogonality relations and operators on bounded quasi-implication algebras" [2507.17724], they arise as the precise algebraic counterparts of quantum monadic algebras, with the correspondence mediated by the Sasaki implication. The theory places monadic quasi-implication algebras at the intersection of implicational algebra, orthomodular lattice theory, and relational semantics via orthoframes.

## 1. Bounded quasi-implication algebras as the underlying implicational structure

The starting point is Hardegree’s notion of a quasi-implication algebra, intended to abstract the behavior of Sasaki implication in orthomodular lattices. A magma $\langle A;\cdot\rangle$ is a quasi-implication algebra when it satisfies the three identities
\[
(x\cdot y)\cdot x=x,
\]
\[
(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),
\]
\[
((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.
\]
From these axioms one obtains
\[
x\cdot(x\cdot y)=x\cdot y,
\qquad
x\cdot x=(x\cdot y)\cdot(x\cdot y),
\qquad
x\cdot x=y\cdot y.
\]
Hence the element
\[
1:=x\cdot x
\]
is well defined, independently of $x$, and satisfies
\[
1\cdot x=x,
\qquad
x\cdot 1=1.
\]

The induced order is defined by
\[
x\preceq y \Longleftrightarrow x\cdot y=1.
\]
A bounded quasi-implication algebra is then a quasi-implication algebra equipped with a constant $0$ such that
\[
0\cdot x=1 \qquad \text{for all }x\in A.
\]
Under the induced order, $0$ is the least element and $1$ is the greatest.

A central structural fact is that every bounded quasi-implication algebra determines an orthomodular lattice via the operations
\[
x^*:=x\cdot 0,
\]
\[
x\oplus y:=((x\cdot y)\cdot(y\cdot x))\cdot x,
\]
\[
x\odot y:=(x^*\oplus y^*)^*.
\]
The equivalent formula
\[
x\odot y=((x\cdot y)\cdot(x\cdot 0))\cdot 0
\]
is recorded as especially useful. Conversely, the Sasaki implication on an orthomodular lattice yields a bounded quasi-implication algebra [2507.17724].

This base theory is essential for the monadic extension. Monadic quasi-implication algebras do not replace the orthomodular lattice structure so much as encode it implicationally, with the lattice operations recoverable from $\cdot$ and $0$.

## 2. Orthogonality, orthoframes, and the pre-monadic relational setting

The theory is motivated in part by generalizing orthogonality constructions due to MacLaren and Goldblatt from ortholattices to bounded quasi-implication algebras. An orthogonality relation on a set $X$ is a binary relation $\perp$ that is irreflexive and symmetric; an orthoframe is a pair $\langle X;\perp\rangle$. For $U\subseteq X$,
\[
U^{\perp}=\{x\in X:x\perp y \text{ for all } y\in U\},
\]
and a subset is bi-orthogonally closed when $U=U^{\perp\perp}$. The family $\mathcal{B}(X)$ of all such subsets forms a complete ortholattice under $\cap$, $\sqcup$, and orthocomplementation.

In the ortholattice setting, MacLaren defined on $A\setminus\{0\}$ the relation
\[
x\perp y \Longleftrightarrow x\le y^\perp,
\]
while Goldblatt defined on the proper filters $\mathfrak{F}(A)$ the relation
\[
\alpha\perp \beta \Longleftrightarrow \exists x\in A \text{ with } x\in\alpha \text{ and } x^\perp\in\beta.
\]

For a bounded quasi-implication algebra $A$, the MacLaren-style relation becomes
\[
x\perp_A^M y \Longleftrightarrow x\cdot (y\cdot 0)=1
\]
on $A\setminus\{0\}$. The resulting structure
\[
X_A^M=\langle A\setminus\{0\};\perp_A^M\rangle
\]
is the MacLaren frame of $A$, and it is proved to be an orthoframe.

The Goldblatt-style construction requires a notion of filter adapted to quasi-implication. A subset $\alpha\subseteq A$ is a filter when:

1. if $x\in \alpha$ and $x\cdot y=1$, then $y\in\alpha$;
2. if $x,y\in\alpha$, then
   \[
   ((x\cdot y)\cdot(x\cdot 0))\cdot 0\in\alpha.
   \]

A filter is proper if $0\notin\alpha$. Writing $\mathfrak{F}(A)$ for the proper filters, one defines
\[
\alpha\perp_A^G \beta \Longleftrightarrow \exists x\in A \text{ such that } x\in \alpha \text{ and } x\cdot 0\in\beta.
\]
Then
\[
X_A^G=\langle \mathfrak{F}(A);\perp_A^G\rangle
\]
is the Goldblatt frame of $A$, again an orthoframe.

These constructions show that bounded quasi-implication algebras support both element-based and filter-based orthogonality semantics. A plausible implication is that the implicational signature already contains enough information to reconstruct substantial parts of the orthogonality-theoretic behavior usually presented in lattice language.

## 3. Quantum monadic algebras and the monadic operator

The monadic extension is formulated first on the ortholattice side. A monadic ortholattice is an algebra
\[
\langle A;\wedge,\vee,^{\perp},0,1,\exists\rangle
\]
with a unary operator $\exists$ satisfying
\[
\exists 0=0,
\qquad
x\le \exists x,
\qquad
\exists(x\vee y)=\exists x\vee\exists y,
\qquad
\exists\exists x=\exists x,
\qquad
\exists(\exists x)^{\perp}=(\exists x)^{\perp}.
\]
A quantum monadic algebra is a monadic ortholattice whose underlying lattice is orthomodular.

The operator $\exists$ is emphasized as a closure operator whose fixed points form an orthomodular sublattice. Dually,
\[
\forall x := (\exists x^\perp)^\perp
\]
defines an interior operator.

This setting is the immediate model for monadic quasi-implication algebras. The latter are designed so that $\Diamond$ plays the role of $\exists$, but in an implicational signature built around $\cdot$ and $0$ rather than around $\wedge$, $\vee$, and orthocomplementation. The monadic behavior is therefore encoded through identities expressed entirely in the quasi-implication language.

The theory also identifies a limitation of direct analogy with the distributive case. A quantum monadic algebra need not satisfy
\[
\exists(x\wedge \exists y)=\exists x\wedge \exists y,
\]
or
\[
\forall(x\vee \forall y)=\forall x\vee \forall y.
\]
Accordingly, the monadic operator in the orthomodular setting is strictly weaker than the familiar behavior in monadic Boolean algebras [2507.17724].

## 4. Definition and internal properties of monadic quasi-implication algebras

A monadic quasi-implication algebra is an algebra
\[
\langle A;\cdot,0,\Diamond\rangle
\]
such that $\langle A;\cdot,0\rangle$ is a bounded quasi-implication algebra and $\Diamond$ satisfies the identities
\[
\Diamond\Diamond x\cdot\Diamond x=1
\quad\text{and}\quad
x\cdot\Diamond x=1,
\]
\[
\Diamond(\Diamond x\cdot 0)=\Diamond x\cdot 0
\quad\text{and}\quad
\Diamond 0=0,
\]
\[
\Diamond(((x\cdot 0)\cdot(y\cdot 0))\cdot x)=((\Diamond x\cdot 0)\cdot(\Diamond y\cdot 0))\cdot \Diamond x.
\]

These axioms are the defining conditions of the concept. Their immediate algebraic consequences mirror the closure-operator behavior of $\exists$ in a quantum monadic algebra. In particular, $\Diamond$ is idempotent:
\[
\Diamond\Diamond x=\Diamond x.
\]
The proof uses the induced order $\preceq$, since $x\cdot\Diamond x=1$ and $\Diamond\Diamond x\cdot\Diamond x=1$ give both $\Diamond\Diamond x\preceq \Diamond x$ and $\Diamond x\preceq \Diamond\Diamond x$.

The paper also proves that $\Diamond$ is monotone with respect to $\preceq$. Together with $\Diamond 0=0$ and $x\preceq \Diamond x$, this places $\Diamond$ in the role of a closure operator internal to the quasi-implication setting. The additive behavior needed for the orthomodular reconstruction is captured not by a lattice equation in primitive form, but by the final defining identity involving
\[
((x\cdot 0)\cdot(y\cdot 0))\cdot x.
\]

This notion is the central new algebraic object of the theory. It is not introduced as an isolated variety; rather, it is constructed specifically so that the monadic orthomodular and implicational formalisms coincide exactly.

## 5. Translation through the Sasaki implication and categorical isomorphism

The bridge between the two settings is the Sasaki implication
\[
p_s(x,y)=x^\perp\vee(x\wedge y).
\]
This is one of the three ortholattice polynomials identified by Hardegree as satisfying the algebraic implicational conditions in orthomodular lattices.

If $A$ is a quantum monadic algebra, then
\[
\langle A;p_s,0,\exists\rangle
\]
is a monadic quasi-implication algebra. The verification uses the fact that $p_s$ makes the orthomodular lattice into a bounded quasi-implication algebra, the quantifier axioms for $\exists$, and the identity
\[
p_s(x,0)=x^\perp.
\]
A key supporting lemma is
\[
p_s(p_s(x^{\perp},y^{\perp}),x)=x\vee y,
\]
which is used in the monadic verification.

Conversely, if $A$ is a monadic quasi-implication algebra, then
\[
\langle A;\odot,\oplus,^*,0,1,\Diamond\rangle
\]
is a quantum monadic algebra, where
\[
x^*:=x\cdot 0,
\qquad
x\oplus y:=((x\cdot y)\cdot(y\cdot x))\cdot x,
\qquad
x\odot y=(x^*\oplus y^*)^*.
\]
The axioms for $\Diamond$ ensure normality, extensivity, additivity,
\[
\Diamond(x\oplus y)=\Diamond x\oplus \Diamond y,
\]
idempotence, and preservation of orthocomplements of closed elements,
\[
\Diamond(\Diamond x)^* = (\Diamond x)^*.
\]

The two translations are mutually inverse on the nose. Starting with a quantum monadic algebra, passing to the monadic quasi-implication algebra and reconstructing returns the original structure. Starting with a monadic quasi-implication algebra, passing to the quantum monadic algebra and then back also returns the original structure. One of the central identities used in the reverse direction is
\[
x^*\oplus(x\odot y)=x\cdot y.
\]

This exact correspondence extends to morphisms. The category $\mathbf{QMA}$ has as objects quantum monadic algebras and as morphisms homomorphisms preserving the bounded lattice operations, orthocomplementation, and $\exists$. The category $\mathbf{MQIA}$ has as objects monadic quasi-implication algebras and as morphisms homomorphisms preserving $\cdot$, $0$, and $\Diamond$. The main categorical theorem is
\[
\mathbf{QMA}\cong \mathbf{MQIA}.
\]
This is an isomorphism of categories, not merely an equivalence in a loose sense. For instance, a $\mathbf{QMA}$-homomorphism satisfies
\[
h(p_s(x,y))=p_s(h(x),h(y)),
\]
and an $\mathbf{MQIA}$-homomorphism satisfies
\[
h(x\odot y)=h(x)\odot h(y)
\]
after reconstruction [2507.17724].

## 6. Monadic orthoframes, examples, and scope of the theory

The relational semantics is extended from orthoframes to monadic orthoframes. A monadic orthoframe is a triple
\[
\langle X;\perp,R\rangle
\]
such that $\langle X;\perp\rangle$ is an orthoframe, $R$ is reflexive and transitive, and for every $x\in X$,
\[
R[R[\{x\}]^{\perp}]\subseteq R[\{x\}]^{\perp}.
\]

Given a monadic quasi-implication algebra $A$, one defines on $A\setminus\{0\}$
\[
xR_A^M y \Longleftrightarrow y\cdot \Diamond x=1
\]
together with the MacLaren orthogonality $\perp_A^M$. Then
\[
X_A^M=\langle A\setminus\{0\};\perp_A^M,R_A^M\rangle
\]
is a monadic orthoframe. Reflexivity follows from $x\cdot\Diamond x=1$, while transitivity follows from monotonicity and idempotence of $\Diamond$. The compatibility condition
\[
R[R[\{x\}]^{\perp}]\subseteq R[\{x\}]^{\perp}
\]
is derived using technical lemmas on orthogonal complements and the identity
\[
\Diamond(\Diamond x\cdot 0)=\Diamond x\cdot 0.
\]

There is also a Goldblatt-style monadic construction on proper filters. Defining
\[
\alpha R_A^G \beta \Longleftrightarrow \Diamond[\alpha]\subseteq \beta,
\qquad
\Diamond[\alpha]=\{\Diamond x : x\in\alpha\},
\]
one obtains
\[
X_A^G=\langle \mathfrak{F}(A);\perp_A^G,R_A^G\rangle,
\]
which is again a monadic orthoframe.

The examples clarify the expressive range of the framework. The paper gives a concrete orthomodular lattice with elements $0,1,x,x^\perp,y,y^\perp$ whose Sasaki implication table yields a quasi-implication algebra that is not a Boolean implication algebra. The witness is the failure of quasi-commutativity:
\[
p_s(p_s(x,y),y)\neq p_s(p_s(y,x),x).
\]
Thus quasi-implication algebras properly generalize Boolean implication algebras. At the opposite extreme, when the ortholattice is distributive, the principal implication operations coincide:
\[
p_c(x,y)=p_s(x,y)=p_d(x,y)=p_k(x,y).
\]

These facts address a common misconception: the monadic quasi-implication framework is neither merely a reformulation of Boolean implication algebras nor a trivial restatement of orthomodular lattice theory. It is a non-distributive implicational formalism with exact algebraic and categorical correspondence to quantum monadic algebras, together with MacLaren-style and Goldblatt-style relational semantics.

A final remark concerns scope. The orthoframe constructions are observed not to use orthomodularity in an essential way, and future work is proposed on Kripke-style frames for quantum monadic algebras. The stated obstacle is that orthomodularity is not first-order definable from orthogonality alone [2507.17724].

Source: https://www.emergentmind.com/topics/monadic-quasi-implication-algebras