Bounded Quasi-Implication Algebras

• Bounded quasi-implication algebras are implication-based structures derived from orthomodular lattices, featuring a constant 0 and a definable top element via Sasaki implication.
• They recover the complete lattice structure by defining order, orthocomplement, join, and meet solely through the application of the implication operation.
• Recent advancements extend the framework with orthogonality relations, monadic operators, and frame constructions that establish a categorical isomorphism with quantum monadic algebras.

Searching arXiv for the cited work and closely related supporting papers. Bounded quasi-implication algebras are implication-based algebras introduced by Hardegree as counterparts of orthomodular lattices. In the formulation studied in “Orthogonality relations and operators on bounded quasi-implication algebras” (McDonald, 23 Jul 2025), a bounded quasi-implication algebra is a quasi-implication algebra equipped with a constant $0$ such that $0\cdot x=1$ for all $x$, where the top element is not primitive but is definable by $1:=x\cdot x$. Their significance lies in the fact that an implication-only signature $\langle 2,0\rangle$ already determines order, orthocomplement, join, and meet, and therefore recovers the orthomodular structure that motivated the formalism. The 2025 development of the subject extends this Hardegree framework in three directions: orthogonality relations, monadic operators, and frame constructions (McDonald, 23 Jul 2025).

1. Orthomodular origin and basic definition

The background setting is the theory of ortholattices and orthomodular lattices. An ortholattice is an algebra

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$

such that $\langle A;\wedge,\vee,0,1\rangle$ is a bounded lattice and $^{\perp}$ is an orthocomplementation satisfying

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$

$$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$

An orthomodular lattice is an ortholattice satisfying

$0\cdot x=1$0

Hardegree’s motivation was to isolate an implication operation $0\cdot x=1$1 on an ortholattice that satisfies algebraic counterparts of implication, modus ponens, and modus tollens: $0\cdot x=1$2

$0\cdot x=1$3

$0\cdot x=1$4

In orthomodular lattices, exactly three ortholattice polynomials satisfy these conditions: Sasaki implication,

$0\cdot x=1$5

Dishkant implication,

$0\cdot x=1$6

and Kalmbach implication,

$0\cdot x=1$7

The bounded quasi-implication algebra formalism is specifically tied to the equational behavior of Sasaki implication (McDonald, 23 Jul 2025).

A quasi-implication algebra is a magma $0\cdot x=1$8 satisfying Hardegree’s three equations: $0\cdot x=1$9

$x$0

$x$1

From these one derives

$x$2

$x$3

$x$4

Hence

$x$5

is well defined, and moreover

$x$6

The induced order is

$x$7

A bounded quasi-implication algebra is a quasi-implication algebra with a constant $x$8 such that

$x$9

Its intended interpretation is that $1:=x\cdot x$0 behaves like Sasaki implication, $1:=x\cdot x$1 is the top element, and $1:=x\cdot x$2 is a bottom element. The induced order $1:=x\cdot x$3 has $1:=x\cdot x$4 as least and $1:=x\cdot x$5 as greatest element (McDonald, 23 Jul 2025).

A common misconception is that quasi-implication algebras are merely Boolean implication algebras in unusual notation. The class is strictly larger. Every Boolean implication algebra is a quasi-implication algebra, but the converse fails. The witness given in the orthomodular setting is failure of quasi-commutativity: $1:=x\cdot x$6 since

$1:=x\cdot x$7

This separates bounded quasi-implication algebras from bounded Boolean implication algebras (McDonald, 23 Jul 2025).

2. Reconstruction of orthomodular structure from implication

A central theorem recalled from Hardegree states that every bounded quasi-implication algebra induces an orthomodular lattice

$1:=x\cdot x$8

where

$1:=x\cdot x$9

$\langle 2,0\rangle$0

$\langle 2,0\rangle$1

The paper also records the derived formula

$\langle 2,0\rangle$2

hence

$\langle 2,0\rangle$3

These identities are structurally decisive because they recover the lattice-theoretic operations from implication-only data (McDonald, 23 Jul 2025).

The induced order, complement, join, and meet are therefore all definable in the quasi-implication language: $\langle 2,0\rangle$4

$\langle 2,0\rangle$5

This shows that the formal weakness of the signature is deceptive. The implication operation is not merely a reductive remnant of orthomodular lattice theory; it determines the full orthomodular lattice structure.

The converse reconstruction also matters. In the round-trip from the induced orthomodular structure back to the implication operation, the key identity is

$\langle 2,0\rangle$6

This exact recovery is what underwrites later equivalence and isomorphism results in the monadic setting. A plausible implication is that bounded quasi-implication algebras should be viewed less as impoverished algebras and more as alternative presentations of orthomodular behavior, with Sasaki implication as the primitive operation.

3. Orthogonality relations and frame constructions

A major development is the transfer of orthogonality-frame constructions from ortholattices to bounded quasi-implication algebras. An orthogonality relation on a set $\langle 2,0\rangle$7 is a binary relation $\langle 2,0\rangle$8 that is irreflexive and symmetric; an orthoframe is $\langle 2,0\rangle$9. For $\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$0,

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$1

and $\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$2 is bi-orthogonally closed if

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$3

The family

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$4

forms a complete ortholattice by Birkhoff’s theorem, with join

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$5

The first construction is the MacLaren orthogonality on nonzero elements. For a bounded quasi-implication algebra $\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$6,

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$7

Since $\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$8, this is the implication-language version of

$\langle A;\wedge,\vee,{}^{\perp},0,1\rangle$9

The associated frame is

$\langle A;\wedge,\vee,0,1\rangle$0

The paper proves that $\langle A;\wedge,\vee,0,1\rangle$1 is an orthoframe. Irreflexivity follows because $\langle A;\wedge,\vee,0,1\rangle$2 implies

$\langle A;\wedge,\vee,0,1\rangle$3

but $\langle A;\wedge,\vee,0,1\rangle$4, so $\langle A;\wedge,\vee,0,1\rangle$5, hence $\langle A;\wedge,\vee,0,1\rangle$6, a contradiction. Symmetry is obtained from Hardegree’s identities, including $\langle A;\wedge,\vee,0,1\rangle$7 (McDonald, 23 Jul 2025).

The second construction is the Goldblatt orthogonality on proper filters. A nonempty $\langle A;\wedge,\vee,0,1\rangle$8 is a filter if it is upward closed under the induced order and closed under the reconstructed meet: $\langle A;\wedge,\vee,0,1\rangle$9 A filter is proper iff $^{\perp}$0, and $^{\perp}$1 denotes the set of proper filters. Goldblatt orthogonality is defined by

$^{\perp}$2

The associated frame is

$^{\perp}$3

The paper proves that $^{\perp}$4 is also an orthoframe. Irreflexivity comes from the fact that if $^{\perp}$5, then filter closure yields

$^{\perp}$6

which simplifies to

$^{\perp}$7

contradicting propriety. Symmetry uses

$^{\perp}$8

These two orthogonality relations reveal two levels of structure. The first is element-level orthogonality on $^{\perp}$9, mirroring orthogonality in the reconstructed orthomodular lattice. The second is filter-level orthogonality on $x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$0, designed for completion and canonical-extension methods. In both cases the novelty is not a new notion of orthogonality, but the re-expression of MacLaren and Goldblatt constructions in the implication-only language of bounded quasi-implication algebras (McDonald, 23 Jul 2025).

4. Monadic enrichment

The monadic extension of the theory introduces a unary operator $x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$1. A monadic quasi-implication algebra is an algebra

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$2

such that $x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$3 is a bounded quasi-implication algebra and $x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$4 satisfies

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$5

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$6

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$7

The notation is deliberately modal and quantificational. The axioms are designed so that, after reconstruction of the orthomodular lattice operations, $x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$8 becomes a quantifier or closure operator (McDonald, 23 Jul 2025).

The order-theoretic interpretation of the axioms is explicit. The equation

$x\wedge x^{\perp}=0,\qquad x\vee x^{\perp}=1,$9

states $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$0, so $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$1 is inflationary. Together with

$$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$2

and antisymmetry of $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$3, one obtains idempotence: $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$4 The identity

$$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$5

states that the orthocomplement of a closed element is closed. The longer axiom is the implication-language translation of additivity over joins.

The motivating comparison is with quantifiers on orthomodular lattices. A quantum monadic algebra is an orthomodular lattice equipped with a closure operator whose closed elements form an orthomodular sublattice. The monadic quasi-implication algebra axioms transport that theory into the implication-only setting. This places bounded quasi-implication algebras in direct contact with the non-distributive analogue of Halmos’s monadic Boolean algebras (McDonald, 23 Jul 2025).

The monadic enrichment also extends the frame theory. A monadic orthoframe is a triple

$$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$6

such that $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$7 is an orthoframe, $$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$8 is reflexive and transitive, and

$$x\le y \Rightarrow y^{\perp}\le x^{\perp},\qquad x=x^{\perp\perp}.$$9

for all $0\cdot x=1$00. For a monadic quasi-implication algebra, the monadic MacLaren relation is

$0\cdot x=1$01

and the monadic Goldblatt relation is

$0\cdot x=1$02

The paper proves that both yield monadic orthoframes. Reflexivity uses $0\cdot x=1$03; transitivity uses monotonicity and idempotence of $0\cdot x=1$04; the closure condition uses the identities

$0\cdot x=1$05

and

$0\cdot x=1$06

These constructions generalize Harding’s monadic MacLaren-style frame and Harding–McDonald–Peinado’s monadic Goldblatt-style frame from monadic ortholattices to monadic quasi-implication algebras (McDonald, 23 Jul 2025).

5. Quantum monadic algebras and categorical isomorphism

The monadic theory is organized around an exact correspondence with quantum monadic algebras. A monadic ortholattice is

$0\cdot x=1$07

such that $0\cdot x=1$08 is an ortholattice and $0\cdot x=1$09 satisfies

$0\cdot x=1$10

$0\cdot x=1$11

$0\cdot x=1$12

$0\cdot x=1$13

A quantum monadic algebra is a monadic ortholattice whose lattice reduct is orthomodular. The dual interior operator is

$0\cdot x=1$14

The first conversion theorem states that if $0\cdot x=1$15 is a quantum monadic algebra, then

$0\cdot x=1$16

is a monadic quasi-implication algebra, where

$0\cdot x=1$17

Two identities drive the translation: $0\cdot x=1$18 and

$0\cdot x=1$19

The latter is what converts the monadic join axiom $0\cdot x=1$20 into the quasi-implication axiom for $0\cdot x=1$21 (McDonald, 23 Jul 2025).

The converse conversion theorem states that if $0\cdot x=1$22 is a monadic quasi-implication algebra, then

$0\cdot x=1$23

is a quantum monadic algebra, where

$0\cdot x=1$24

The proof shows that $0\cdot x=1$25 is a quantifier: it is normal, inflationary, additive, idempotent, and preserves complements of closed elements.

The paper then proves round-trip results on objects. Starting from a quantum monadic algebra, passing to the Sasaki implication structure and then reconstructing the orthomodular operations returns the original algebra. Starting from a monadic quasi-implication algebra, passing to the induced quantum monadic algebra and then back via Sasaki implication yields the original implication structure. The key formula for the second direction is again

$0\cdot x=1$26

At the categorical level, the paper defines $0\cdot x=1$27 as the category of quantum monadic algebras and $0\cdot x=1$28 as the category of monadic quasi-implication algebras. The main theorem states

$0\cdot x=1$29

On objects, the correspondence is

$0\cdot x=1$30

and

$0\cdot x=1$31

On morphisms, the underlying function is unchanged, and preservation of the nonprimitive operations follows from definability. The result is stronger than a loose correspondence: the conversions are literally inverse definitional expansions and reductions (McDonald, 23 Jul 2025).

6. Position within implication-based algebra and current limits

Bounded quasi-implication algebras occupy a distinctive position among implication-centered algebraic systems. They are not the same as bounded implication algebras in Abbott’s classical sense. In “Monadic and cylindric expansions of bounded implication algebras,” bounded implication algebras are implication algebras with a constant $0\cdot x=1$32 such that $0\cdot x=1$33, and every such algebra can be converted into a Boolean algebra with

$0\cdot x=1$34

yielding the categorical isomorphism

$0\cdot x=1$35

By contrast, bounded quasi-implication algebras are implication-based counterparts of orthomodular lattices and properly generalize Boolean implication algebras rather than collapsing to them (McDonald, 2 Jun 2026).

They are also distinct from the broader residuated and substructural setting of generalized BI-algebras. “An Algebraic Glimpse at Bunched Implications and Separation Logic” does not define bounded quasi-implication algebras, but studies Heyting algebras, GBI-algebras, BI-algebras, Boolean BI-algebras, and related bounded implicative-residuated structures. In that framework, the closest exact neighbors are Heyting algebras and the subvariety $0\cdot x=1$36 defined by

$0\cdot x=1$37

while the weakening or integral subvariety $0\cdot x=1$38 is characterized by

$0\cdot x=1$39

This suggests that bounded quasi-implication algebras belong to a wider landscape of implication-dominant algebras, but their orthomodular origin sharply differentiates them from distributive residuated systems (Jipsen et al., 2017).

A further nearby development is the theory of Johnstone algebras. “Semi-abelian by Design: Johnstone Algebras Unifying Implication and Division” does not define bounded quasi-implication algebras, but it studies implication-based order

$0\cdot x=1$40

and the relative closure term

$0\cdot x=1$41

Its M-axiom,

$0\cdot x=1$42

is presented as a weakening of the H-axiom to comparable elements. A plausible implication is that closure-term methods of this kind may be useful in other implication-only settings, although no such transfer is proved for bounded quasi-implication algebras in that paper (Forsman, 28 Apr 2025).

The current limit of the bounded quasi-implication program concerns relational semantics. The conclusion of the 2025 note states that the monadic orthoframe constructions do not exploit orthomodularity of the induced lattice. This motivates the search for Kripke frames for quantum monadic algebras, namely relational structures whose bi-orthogonally closed subsets form genuine quantum monadic algebras. The obstacle is that orthomodularity is not first-order definable by orthogonality relations alone, and Goldblatt’s result that orthomodularity is not elementary is cited as the central obstruction. The resulting research direction is therefore not merely to refine existing orthoframes, but to strengthen the relational semantics beyond orthogonality alone (McDonald, 23 Jul 2025).