Monadic Quasi-Implication Algebras

• Monadic quasi-implication algebras are bounded quasi-implication algebras enhanced with a monadic operator that mirrors quantum monadic algebra features.
• They use the Sasaki implication to recover orthomodular lattice properties and establish a categorical isomorphism with quantum monadic algebras.
• The framework integrates both MacLaren-style element-based and Goldblatt-style filter-based orthoframe semantics for non-distributive quantum logics.

Searching arXiv for the target paper and closely related work on quasi-implication algebras and monadic/orthomodular structures. Tool unavailable in this interface, so I will rely on the provided arXiv metadata and the supplied paper summary, citing the target paper directly as (McDonald, 23 Jul 2025). 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" (McDonald, 23 Jul 2025), 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 $\langle A;\cdot,0,\Diamond\rangle$0, and satisfies

$\langle A;\cdot,0,\Diamond\rangle$1

The induced order is defined by

$\langle A;\cdot,0,\Diamond\rangle$2

A bounded quasi-implication algebra is then a quasi-implication algebra equipped with a constant $\langle A;\cdot,0,\Diamond\rangle$3 such that

$\langle A;\cdot,0,\Diamond\rangle$4

Under the induced order, $\langle A;\cdot,0,\Diamond\rangle$5 is the least element and $\langle A;\cdot,0,\Diamond\rangle$6 is the greatest.

A central structural fact is that every bounded quasi-implication algebra determines an orthomodular lattice via the operations

$\langle A;\cdot,0,\Diamond\rangle$7

$\langle A;\cdot,0,\Diamond\rangle$8

$\langle A;\cdot,0,\Diamond\rangle$9

The equivalent formula

$\cdot$0

is recorded as especially useful. Conversely, the Sasaki implication on an orthomodular lattice yields a bounded quasi-implication algebra (McDonald, 23 Jul 2025).

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$1 and $\cdot$2.

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 $\cdot$3 is a binary relation $\cdot$4 that is irreflexive and symmetric; an orthoframe is a pair $\cdot$5. For $\cdot$6,

$\cdot$7

and a subset is bi-orthogonally closed when $\cdot$8. The family $\cdot$9 of all such subsets forms a complete ortholattice under $\Diamond$0, $\Diamond$1, and orthocomplementation.

In the ortholattice setting, MacLaren defined on $\Diamond$2 the relation

$\Diamond$3

while Goldblatt defined on the proper filters $\Diamond$4 the relation

$\Diamond$5

For a bounded quasi-implication algebra $\Diamond$6, the MacLaren-style relation becomes

$\Diamond$7

on $\Diamond$8. The resulting structure

$\Diamond$9

is the MacLaren frame of $\langle A;\cdot\rangle$0, and it is proved to be an orthoframe.

The Goldblatt-style construction requires a notion of filter adapted to quasi-implication. A subset $\langle A;\cdot\rangle$1 is a filter when:

1. if $\langle A;\cdot\rangle$2 and $\langle A;\cdot\rangle$3, then $\langle A;\cdot\rangle$4;
2. if $\langle A;\cdot\rangle$5, then

$\langle A;\cdot\rangle$6

A filter is proper if $\langle A;\cdot\rangle$7. Writing $\langle A;\cdot\rangle$8 for the proper filters, one defines

$\langle A;\cdot\rangle$9

Then

$(x\cdot y)\cdot x=x,$0

is the Goldblatt frame of $(x\cdot y)\cdot x=x,$1, 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

$(x\cdot y)\cdot x=x,$2

with a unary operator $(x\cdot y)\cdot x=x,$3 satisfying

$(x\cdot y)\cdot x=x,$4

A quantum monadic algebra is a monadic ortholattice whose underlying lattice is orthomodular.

The operator $(x\cdot y)\cdot x=x,$5 is emphasized as a closure operator whose fixed points form an orthomodular sublattice. Dually,

$(x\cdot y)\cdot x=x,$6

defines an interior operator.

This setting is the immediate model for monadic quasi-implication algebras. The latter are designed so that $(x\cdot y)\cdot x=x,$7 plays the role of $(x\cdot y)\cdot x=x,$8, but in an implicational signature built around $(x\cdot y)\cdot x=x,$9 and $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$0 rather than around $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$1, $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$2, 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

$$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$3

or

$$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$4

Accordingly, the monadic operator in the orthomodular setting is strictly weaker than the familiar behavior in monadic Boolean algebras (McDonald, 23 Jul 2025).

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

A monadic quasi-implication algebra is an algebra

$$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$5

such that $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$6 is a bounded quasi-implication algebra and $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$7 satisfies the identities

$$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$8

$$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$$9

$$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$0

These axioms are the defining conditions of the concept. Their immediate algebraic consequences mirror the closure-operator behavior of $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$1 in a quantum monadic algebra. In particular, $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$2 is idempotent: $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$3 The proof uses the induced order $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$4, since $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$5 and $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$6 give both $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$7 and $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$8.

The paper also proves that $$((x\cdot y)\cdot(y\cdot x))\cdot x=((y\cdot x)\cdot(x\cdot y))\cdot y.$$9 is monotone with respect to $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$0. Together with $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$1 and $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$2, this places $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$3 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

$$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$4

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

$$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$5

This is one of the three ortholattice polynomials identified by Hardegree as satisfying the algebraic implicational conditions in orthomodular lattices.

If $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$6 is a quantum monadic algebra, then

$$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$7

is a monadic quasi-implication algebra. The verification uses the fact that $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$8 makes the orthomodular lattice into a bounded quasi-implication algebra, the quantifier axioms for $$x\cdot(x\cdot y)=x\cdot y, \qquad x\cdot x=(x\cdot y)\cdot(x\cdot y), \qquad x\cdot x=y\cdot y.$$9, and the identity

$1:=x\cdot x$0

A key supporting lemma is

$1:=x\cdot x$1

which is used in the monadic verification.

Conversely, if $1:=x\cdot x$2 is a monadic quasi-implication algebra, then

$1:=x\cdot x$3

is a quantum monadic algebra, where

$1:=x\cdot x$4

The axioms for $1:=x\cdot x$5 ensure normality, extensivity, additivity,

$1:=x\cdot x$6

idempotence, and preservation of orthocomplements of closed elements,

$1:=x\cdot x$7

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

$1:=x\cdot x$8

This exact correspondence extends to morphisms. The category $1:=x\cdot x$9 has as objects quantum monadic algebras and as morphisms homomorphisms preserving the bounded lattice operations, orthocomplementation, and $\langle A;\cdot,0,\Diamond\rangle$00. The category $\langle A;\cdot,0,\Diamond\rangle$01 has as objects monadic quasi-implication algebras and as morphisms homomorphisms preserving $\langle A;\cdot,0,\Diamond\rangle$02, $\langle A;\cdot,0,\Diamond\rangle$03, and $\langle A;\cdot,0,\Diamond\rangle$04. The main categorical theorem is

$\langle A;\cdot,0,\Diamond\rangle$05

This is an isomorphism of categories, not merely an equivalence in a loose sense. For instance, a $\langle A;\cdot,0,\Diamond\rangle$06-homomorphism satisfies

$\langle A;\cdot,0,\Diamond\rangle$07

and an $\langle A;\cdot,0,\Diamond\rangle$08-homomorphism satisfies

$\langle A;\cdot,0,\Diamond\rangle$09

after reconstruction (McDonald, 23 Jul 2025).

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 A;\cdot,0,\Diamond\rangle$10

such that $\langle A;\cdot,0,\Diamond\rangle$11 is an orthoframe, $\langle A;\cdot,0,\Diamond\rangle$12 is reflexive and transitive, and for every $\langle A;\cdot,0,\Diamond\rangle$13,

$\langle A;\cdot,0,\Diamond\rangle$14

Given a monadic quasi-implication algebra $\langle A;\cdot,0,\Diamond\rangle$15, one defines on $\langle A;\cdot,0,\Diamond\rangle$16

$\langle A;\cdot,0,\Diamond\rangle$17

together with the MacLaren orthogonality $\langle A;\cdot,0,\Diamond\rangle$18. Then

$\langle A;\cdot,0,\Diamond\rangle$19

is a monadic orthoframe. Reflexivity follows from $\langle A;\cdot,0,\Diamond\rangle$20, while transitivity follows from monotonicity and idempotence of $\langle A;\cdot,0,\Diamond\rangle$21. The compatibility condition

$\langle A;\cdot,0,\Diamond\rangle$22

is derived using technical lemmas on orthogonal complements and the identity

$\langle A;\cdot,0,\Diamond\rangle$23

There is also a Goldblatt-style monadic construction on proper filters. Defining

$\langle A;\cdot,0,\Diamond\rangle$24

one obtains

$\langle A;\cdot,0,\Diamond\rangle$25

which is again a monadic orthoframe.

The examples clarify the expressive range of the framework. The paper gives a concrete orthomodular lattice with elements $\langle A;\cdot,0,\Diamond\rangle$26 whose Sasaki implication table yields a quasi-implication algebra that is not a Boolean implication algebra. The witness is the failure of quasi-commutativity: $\langle A;\cdot,0,\Diamond\rangle$27 Thus quasi-implication algebras properly generalize Boolean implication algebras. At the opposite extreme, when the ortholattice is distributive, the principal implication operations coincide: $\langle A;\cdot,0,\Diamond\rangle$28

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 (McDonald, 23 Jul 2025).