Conditional Esakia Spaces

• Conditional Esakia spaces are a dual topological semantics for intuitionistic conditional logic, characterized by Esakia spaces enriched with clopen-indexed relational structures.
• They preserve the framework of conditional Heyting algebras by refining standard Esakia spaces with relations defined solely on clopen upsets.
• They enable the transfer of algebraic completeness to both topological and Kripke completeness through fill-ins that extend relational semantics for non-clopen upsets.

Conditional Esakia spaces are the dual topological semantics for intuitionistic conditional logic: they are Esakia spaces equipped with a family of conditional relations indexed by clopen upsets, so that algebraic completeness for conditional Heyting algebras can be transferred to topological completeness and then, via fill-ins, to Kripke completeness (Dufty et al., 16 Aug 2025). They are a topologised refinement of conditional Kripke frames, but they should not be conflated with other extensions of Esakia theory, such as generalized Esakia spaces among stably locally compact spaces or locally Esakia spaces from order-compactification theory (Hofmann et al., 2014, Almeida et al., 26 Dec 2025).

1. Esakia-theoretic background

An Esakia space is, in the standard setting, a triple $(X,\le,\tau)$ where $(X,\le)$ is a nonempty partial order and $\tau$ is a compact topology such that Priestley separation holds and, for every clopen $a$, the down-closure $\downarrow a$ is clopen (Dufty et al., 16 Aug 2025). Equivalently, a Priestley space is Esakia iff the down-closure of every open set is open; and if $X$ is viewed as a spectral space with patch space $X^p$, then $X$ is Esakia iff for every open $A$ of $X^p$, the down-closure $(X,\le)$0 is open (Hofmann et al., 2014).

This condition is the topological core of intuitionistic duality. In an Esakia space, the lattice $(X,\le)$1 of clopen upsets is a Heyting algebra, and for $(X,\le)$2 the implication is given by

$(X,\le)$3

Conversely, a Priestley space is Esakia iff $(X,\le)$4 is a Heyting algebra (Bezhanishvili et al., 2023). The clopen upsets are therefore not merely topological subsets; they are the dual representatives of algebraic propositions.

This background is decisive for the conditional case. Conditional Esakia spaces do not replace Esakia spaces; they enrich them by adding relational structure compatible with the clopen-upset semantics already used in ordinary Esakia duality.

2. Conditional Kripke frames and conditional Heyting algebras

The conditional language $(X,\le)$5 extends intuitionistic propositional logic by a binary conditional operator $(X,\le)$6. On the frame side, a conditional Kripke frame is a structure

$(X,\le)$7

where $(X,\le)$8 is an intuitionistic Kripke frame and

$(X,\le)$9

is a family of binary relations indexed by all upsets, satisfying the coherence condition

$\tau$0

for every upset $\tau$1 (Dufty et al., 16 Aug 2025).

The semantic clause for the conditional is

$\tau$2

where $\tau$3 is the truth set of $\tau$4. A key proposition states that in any such model every formula denotes an upset (Dufty et al., 16 Aug 2025). This preserves the monotonicity characteristic of intuitionistic semantics.

The algebraic semantics is given by conditional Heyting algebras

$\tau$5

where the additional operation satisfies

$\tau$6

Thus $\tau$7 preserves finite meets in its second argument (Dufty et al., 16 Aug 2025). The corresponding completeness statement is algebraic: $\tau$8 Conditional Esakia spaces arise precisely as the dual topological objects needed to transport this algebraic completeness into a spatial semantics.

3. Definition and internal structure of conditional Esakia spaces

A conditional Esakia space is an Esakia space

$\tau$9

equipped with a family of relations

$a$0

indexed only by clopen upsets, such that three conditions hold (Dufty et al., 16 Aug 2025).

First, for all $a$1,

$a$2

is clopen. This is the dual definability condition for the conditional operator.

Second, for every $a$3,

$a$4

This is the order-stability condition, and it is the topological analogue of the coherence condition for conditional Kripke frames.

Third, for every $a$5 and $a$6, the set $a$7 is closed. The closedness of each fibre is what the cited work uses for duality and compactness arguments.

The restriction to clopen upsets is essential. Clopen upsets are the topological surrogates for algebra elements, so the family $a$8 is defined only where the dual algebraic operation is canonically represented. The resulting structure can be viewed as a kind of general frame with admissible sets $a$9 (Dufty et al., 16 Aug 2025).

A common misconception is that forgetting the topology of a conditional Esakia space automatically yields a full conditional Kripke frame. It does not. A conditional Kripke frame has relations $\downarrow a$0 for all upsets $\downarrow a$1, whereas a conditional Esakia space carries relations only for clopen upsets. The missing relations for non-clopen upsets are exactly the semantic gap that motivates fill-ins (Dufty et al., 16 Aug 2025).

4. Duality and completeness

The duality strategy for intuitionistic conditional logic has three layers. First, completeness is established algebraically for conditional Heyting algebras. Second, this is transferred to spaces via a dual equivalence

$\downarrow a$2

where $\downarrow a$3 is the category of conditional Esakia spaces and conditional bounded morphisms. Third, completeness on spaces is turned into completeness on frames by extending the clopen-indexed relations to all upsets through fill-ins (Dufty et al., 16 Aug 2025).

The functors implementing the duality are explicit. The functor

$\downarrow a$4

sends a conditional Esakia space to its Heyting algebra of clopen upsets, while

$\downarrow a$5

sends a conditional Heyting algebra to its prime-filter Esakia space equipped with the induced conditional relations (Dufty et al., 16 Aug 2025). This extends the classical duality

$\downarrow a$6

between Heyting algebras and Esakia spaces (Bezhanishvili et al., 2023).

The semantic completeness result on the space side is correspondingly direct: for any set of axioms $\downarrow a$7, the logic $\downarrow a$8 is sound and complete with respect to the class of conditional Esakia spaces validating $\downarrow a$9 (Dufty et al., 16 Aug 2025).

General frames provide an intermediate representation. For a general frame $X$0, where $X$1 is the collection of admissible upsets, the complex algebra is

$X$2

A formula is valid on a general frame iff it is valid in its complex algebra (Dufty et al., 16 Aug 2025). In the conditional Esakia setting, the admissible sets are exactly the clopen upsets, so the topological duality is tightly synchronized with the algebraic semantics.

5. Fill-ins, correspondence, and Kripke completeness

A fill-in of a conditional Esakia space

$X$3

is a conditional Kripke frame

$X$4

such that

$X$5

It fills in the missing relations for non-clopen upsets while leaving the clopen-indexed semantics unchanged (Dufty et al., 16 Aug 2025).

The crucial lemma is preservation of falsification: if $X$6 is a fill-in of $X$7, then any formula falsified on $X$8 is also falsified on $X$9. The reason is that clopen valuations on $X^p$0 are still valuations in the fill-in (Dufty et al., 16 Aug 2025). This is the bridge from dual-topological countermodels to ordinary Kripke countermodels.

The paper introduces several fill-ins, each designed for different axiom classes.

Fill-in	Definition for non-clopen $X^p$1	Note
$X^p$2	$X^p$3	Basic fill-in
$X^p$4	$X^p$5	Reflexive style
$X^p$6	$X^p$7	Principal style
$X^p$8	$X^p$9	Total style
$X$0	$X$1	Union fill-in
$X$2	$X$3	Transitive fill-in

The empty fill-in $X$4 already yields completeness for the basic logic: one obtains a conditional Esakia countermodel by duality, extends it by empty relations on non-clopen upsets, and the falsified formula remains falsified in the resulting frame (Dufty et al., 16 Aug 2025).

For axiomatic extensions, the governing notion is persistence. An axiom is $X$5-persistent if whenever it holds on a conditional Esakia space, it still holds after applying the fill-in $X$6. The cited work combines persistence with correspondence theory through a three-step pattern: derive a frame correspondent for an axiom, derive a space correspondent over clopen upsets, and prove that the chosen fill-in transforms spaces satisfying the space-correspondent into frames satisfying the frame-correspondent (Dufty et al., 16 Aug 2025).

The frame correspondents exhibited include the following: $X$7

$X$8

$X$9

$A$0

$A$1

$A$2

Not every axiom is persistent under every fill-in. For example, $A$3 is persistent for $A$4, while $A$5 is persistent for $A$6 (Dufty et al., 16 Aug 2025).

A special case is the squeeze fill-in $A$7, introduced for cautious logics. It is tailored to cautious transitivity, cautious monotonicity, and reflexivity. The central squeeze-related condition is

$A$8

in the reflexive Esakia setting, and it is used to prove completeness for logics such as $A$9 and its extensions (Dufty et al., 16 Aug 2025).

6. Relation to generalized, local, and frame-theoretic Esakia semantics

Conditional Esakia spaces belong to a broader landscape of Esakia-style generalizations, but they solve a different problem from the constructions studied elsewhere.

Generalized Esakia spaces, in the sense of stably locally compact topology, are defined by the same patch-open down-closure condition: $X^p$0 For a stably locally compact space $X^p$1, this is equivalent to the canonical map

$X^p$2

being downwards open, equivalent again to the associated spectral distributor having a right adjoint, and equivalent to $X^p$3 being a split subobject of a compact Hausdorff space in $X^p$4 (Hofmann et al., 2014). In that framework, generalized Esakia spaces and spectral distributors form the idempotent split completion of $X^p$5. This is a categorical extension of ordinary Esakia duality, but it is not a semantics for the conditional connective $X^p$6.

Locally Esakia spaces arise from order-compactification theory. A space is locally Esakia iff it is E-order-zero-dimensional and image-compact, and for order-zero-dimensional $X^p$7 this is equivalent to the existence of an Esakia order-compactification $X^p$8 such that $X^p$9 is an upset of $(X,\le)$00, equivalently to $(X,\le)$01 being such an Esakia order-compactification (Almeida et al., 26 Dec 2025). The functor

$(X,\le)$02

is left adjoint to the inclusion $(X,\le)$03. Again, the issue here is reflective compactification, not conditional semantics.

A third adjacent viewpoint is frame-theoretic. Heyting frames were introduced precisely so that

$(X,\le)$04

thereby recasting Esakia duality through coherent and algebraic frames (Bezhanishvili et al., 2023). This suggests a natural meta-level interpretation: conditional Esakia spaces are best understood not as another topological relaxation of the Esakia condition, but as a semantic enrichment of ordinary Esakia spaces by clopen-indexed residual relations tailored to intuitionistic conditional logic.

Other semantic bridges confirm the centrality of ordinary Esakia spaces in the background. Intuitionistic topological systems, for example, were shown to satisfy

$(X,\le)$05

with the implication clause already formulated in relational Kripke style (Nola et al., 2018). Conditional Esakia spaces continue this pattern at the level of a genuine binary conditional, preserving the Esakia topology while adding exactly the extra relational structure needed for duality, correspondence, and completeness (Dufty et al., 16 Aug 2025).