McKinsey–Tarski Algebras Overview

• McKinsey–Tarski algebras are complete Boolean algebras with an interior operator that meets Kuratowski’s axioms, providing an algebraic framework for topological concepts.
• They bridge classical point-set topology with pointfree topology by structuring open elements into frames and Heyting algebras, facilitating the study of separation and continuity.
• Recent research emphasizes the sensitivity of morphism definitions and categorical obstructions in MT-algebras, influencing duality theories and the development of Raney extensions.

McKinsey–Tarski algebras are algebraic structures arising from the algebraization of topology initiated by McKinsey and Tarski. In the contemporary pointfree-topological literature, an MT-algebra is a complete Boolean algebra equipped with an interior operator satisfying the Kuratowski axioms; dually, one may work with a complete closure algebra, or more generally a complete Boolean algebra with an operator (BAO) satisfying the closure-algebra conditions (Bezhanishvili et al., 2023). The subject now spans several intertwined programs: algebraic models of topological spaces, frame-theoretic and pointfree topology, duality theory, separation axioms, local compactness, and categorical questions about morphisms and colimits. Recent work has also shown that the categorical behavior of MT-algebras depends sharply on the chosen morphism notion: natural MT categories can support dualities with frames and Raney extensions, but can also fail to be cocomplete and therefore fail to be equivalent to varieties (Abbadini et al., 10 Jul 2025).

1. Classical definition and dual formulations

The starting point is the classical notion of an interior algebra. If $B$ is a Boolean algebra, an interior operator is a unary map $\square:B\to B$ satisfying

$$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$

A recent terminological convention defines a McKinsey–Tarski algebra, or MT-algebra, to be an interior algebra $(B,\square)$ whose underlying Boolean algebra is complete (Bezhanishvili et al., 2023). The dual closure operator is

$\Diamond a:=\neg \square\neg a,$

and the same structures can be presented dually as complete closure algebras.

In closure notation, the basic object is a Boolean algebra with operator $(B,C)$ satisfying

$$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$

This is the classical closure-algebra formulation. In the BAO language used in recent category-theoretic work, a BAO is a pair $(B,\Diamond)$ in which $\Diamond$ preserves finite joins; a closure algebra is then characterized by

$$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$

The corresponding MT-algebras in that setting are the complete closure algebras, with morphisms taken to be complete stable morphisms (Abbadini et al., 10 Jul 2025).

The prototypical example is topological. For a space $\square:B\to B$0, the powerset Boolean algebra $\square:B\to B$1 becomes an MT-algebra under interior, and dually a closure algebra under topological closure. Open elements are the fixed points of $\square:B\to B$2, closed elements are the fixed points of the dual closure operator, and the topological interpretation remains the organizing intuition throughout the theory (Bezhanishvili et al., 2023).

2. Pointfree-topological interpretation

An MT-algebra $\square:B\to B$3 carries two canonical substructures. Its open elements

$\square:B\to B$4

form a frame, while its closed elements form a coframe. More precisely, the open part is not merely a bounded sublattice: it is a Heyting algebra, with implication given by

$\square:B\to B$5

This is one of the central bridges between MT-algebras and pointfree topology (Bezhanishvili et al., 2023).

Topological spaces map contravariantly into MT-algebras. A continuous map $\square:B\to B$6 induces the inverse-image homomorphism

$\square:B\to B$7

which is a complete Boolean homomorphism satisfying the MT-morphism condition

$\square:B\to B$8

in the abstract setting. This yields a contravariant functor $\square:B\to B$9. Conversely, passing from an MT-algebra to its frame of opens defines a functor

$$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$0

That functor is essentially surjective: every frame occurs, up to isomorphism, as the open part of some MT-algebra. The construction proceeds through the Boolean envelope of the frame and its MacNeille completion; in related work, this canonical MT-algebra is also described as the Funayama envelope (Bezhanishvili et al., 2023).

The relation with ordinary point-set topology is mediated by atoms. For an MT-algebra $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$1, let $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$2 be the set of atoms of its complete Boolean algebra, and define

$$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$3

The image of $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$4 under $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$5 is a topology on $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$6, producing a contravariant functor $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$7. The resulting pair $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$8 forms a dual adjunction, and it restricts to a dual equivalence between topological spaces and spatial MT-algebras, where “spatial” means “atomic” (Bezhanishvili et al., 2023). This is a decisive difference from ordinary frame theory: frames capture sober spaces, whereas MT-algebras recover all topological spaces on the spatial side.

3. Separation, sobriety, and spatiality

A major development in the recent theory is the internal reformulation of separation axioms. Besides open and closed elements, one uses saturated elements, defined as meets of open elements; locally closed elements, defined as meets of an open and a closed element; and weakly locally closed elements, defined as meets of a saturated and a closed element. These classes supply the generators used to express the MT analogues of $$\square 1=1,\qquad \square(a\wedge b)=\square a\wedge \square b,\qquad \square a\le a,\qquad \square a\le \square\square a.$$9, $(B,\square)$0, $(B,\square)$1, Hausdorff, regular, completely regular, and normal conditions (Bezhanishvili et al., 2023).

The $(B,\square)$2-condition is expressed by requiring weakly locally closed elements to join-generate the algebra. The $(B,\square)$3-condition requires locally closed elements to join-generate it, while the $(B,\square)$4-condition requires closed elements to join-generate it. Hausdorffness is formulated using approximation by regular closed elements; regularity uses the relation

$(B,\square)$5

which restricts on opens to the frame-theoretic “rather below” relation. Complete regularity strengthens this to a rationally interpolated relation $(B,\square)$6, and normality is formulated by separation of disjoint closed elements by disjoint opens. These definitions are calibrated so that for each classical separation level $(B,\square)$7, a topological space $(B,\square)$8 satisfies $(B,\square)$9 exactly when its powerset MT-algebra $\Diamond a:=\neg \square\neg a,$0 satisfies the corresponding MT condition (Bezhanishvili et al., 2023).

Sobriety is imported through a pointfree generation-and-representation condition. One formulation used in the local-compactness theory declares $\Diamond a:=\neg \square\neg a,$1 sober when it is a $\Diamond a:=\neg \square\neg a,$2-algebra and every join-irreducible closed element is the closure of an atom. Under this hypothesis, the comparison map

$\Diamond a:=\neg \square\neg a,$3

is a homeomorphism (Bezhanishvili et al., 2024). Thus sobriety is precisely the condition under which the atom space of the MT-algebra and the point space of its frame of opens coincide.

Spatiality is especially simple in the MT setting: an MT-algebra is spatial if and only if it is atomic. If $\Diamond a:=\neg \square\neg a,$4 is spatial, then the canonical map $\Diamond a:=\neg \square\neg a,$5 is an isomorphism onto the powerset MT-algebra of its atom space. The open frame $\Diamond a:=\neg \square\neg a,$6 is then spatial as well. The converse fails in general: the open frame may be spatial while the MT-algebra is not (Bezhanishvili et al., 3 Aug 2025).

4. Local compactness and duality theory

Local compactness in MT-algebras is formulated internally by means of compact elements and compact containment. An element $\Diamond a:=\neg \square\neg a,$7 is compact if for every family $\Diamond a:=\neg \square\neg a,$8,

$\Diamond a:=\neg \square\neg a,$9

for some finite $(B,C)$0. One writes $(B,C)$1 when there exists a compact element $(B,C)$2 such that

$(B,C)$3

Then $(B,C)$4 is locally compact if for every open $(B,C)$5,

$(B,C)$6

On the open frame, this relation interacts with the usual way-below relation $(B,C)$7: if $(B,C)$8, then $(B,C)$9, and if $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$0 is locally compact, then for open $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$1,

$$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$2

A central theorem states that local compactness of $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$3 implies that $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$4 is a continuous frame, and conversely, if $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$5 is sober and $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$6 is continuous, then $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$7 is locally compact (Bezhanishvili et al., 2024).

The sober case supports an MT version of the Hofmann–Mislove theorem. Open filters of $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$8 are precisely the filters of $$C(a\vee b)=Ca\vee Cb,\qquad C0=0,\qquad a\le Ca=CCa.$$9, and Scott-open filters correspond contravariantly to compact saturated elements: $(B,\Diamond)$0 The correspondence is given by

$(B,\Diamond)$1

with inverse $(B,\Diamond)$2. This theorem is the key technical input for the MT versions of Hofmann–Lawson, Isbell, and Stone dualities developed for locally compact sober MT-algebras, stably locally compact MT-algebras, compact Hausdorff MT-algebras, and zero-dimensional or Stone MT-algebras (Bezhanishvili et al., 2024).

Later work refines the relation between local compactness and spatiality. It had been natural to ask whether the classical frame-theoretic theorem “locally compact implies spatial” extends from frames to MT-algebras. The answer is negative: there exists a locally compact sober MT-algebra that is not spatial. The construction uses Raney extensions and the coframe of strongly exact filters $(B,\Diamond)$3 over a locally compact Hausdorff dense-in-itself space such as $(B,\Diamond)$4, and shows that the dense-open filter $(B,\Diamond)$5 is strongly exact but contained in no completely prime filter (Bezhanishvili et al., 3 Aug 2025). At the same time, a positive theorem survives at the $(B,\Diamond)$6-level: every locally compact $(B,\Diamond)$7-algebra is spatial. The proof uses the existence of closed atoms in compact $(B,\Diamond)$8-algebras, and the paper shows that this closed-atom principle is equivalent to the axiom of choice (Bezhanishvili et al., 3 Aug 2025).

5. Morphisms, Raney extensions, and categorical obstructions

The theory of MT-algebras is unusually sensitive to the choice of morphisms. In the pointfree-topological development, an MT-morphism is a complete Boolean homomorphism preserving interior only up to the inequality

$(B,\Diamond)$9

while in the BAO presentation the dual condition is stability,

$\Diamond$0

Additional morphism notions have been introduced to recover stronger categorical correspondences. Proximity morphisms yield a category of MT-algebras equivalent to the category of frames, and recent work introduces Raney morphisms as the correct morphisms for relating MT-algebras to Raney extensions (Bezhanishvili et al., 1 Sep 2025).

A Raney extension is a pair $\Diamond$1 where $\Diamond$2 is a coframe, $\Diamond$3 is a subframe, $\Diamond$4 meet-generates $\Diamond$5, and

$\Diamond$6

for every $\Diamond$7 and $\Diamond$8. Every MT-algebra $\Diamond$9 gives such a pair

$$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$0

where $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$1 is the coframe of saturated elements and $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$2 is the frame of opens. In this language, $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$3 is a $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$4-algebra exactly when $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$5 join-generates $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$6. A Raney morphism $$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$7 acts as a coframe morphism on saturated elements and a frame morphism on opens, preserves finite meets, preserves joins on saturated elements, and satisfies the approximation formula

$$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$8

With composition defined by

$$a\le \Diamond a,\qquad \Diamond\Diamond a\le \Diamond a.$$9

the resulting category $\square:B\to B$00 is equivalent to the category of Raney extensions. The generalized Funayama envelope then yields the $\square:B\to B$01-hull of a Raney extension, extending the older Funayama-envelope/$\square:B\to B$02-hull construction for frames (Bezhanishvili et al., 1 Sep 2025).

A very different picture emerges in the BAO/stable-morphism setting. There, the category of MT-algebras and MT-morphisms is not cocomplete: it lacks some countable copowers, and therefore it is not equivalent to a prevariety, let alone a variety. This answers a question of Peter Jipsen in the negative (Abbadini et al., 10 Jul 2025). The proof first shows that if a colimit-preserving functor $\square:B\to B$03 sees an object whose Boolean reduct is the four-element Boolean algebra, then $\square:B\to B$04 cannot have all countable copowers, because such copowers would force the existence of a free countable Boolean algebra, known not to exist by results of Gaifman and Hales. The argument is then applied to complete BAO categories by constructing left and right adjoints to the forgetful functor via the operators

$\square:B\to B$05

The four-element Boolean algebra with $\square:B\to B$06 lies in the MT class, so the obstruction applies. A duality-theoretic proof supplements this: on the dual Stone-relational side, equalizers can fail because the restricted relation ceases to be continuous. This suggests that the categorical behavior of MT-algebras is controlled at least as much by the morphism notion as by the object class itself.

6. Generalizations and related programs

One important extension is the theory of tangled closure algebras. Here the unary closure operator is supplemented by an operation

$\square:B\to B$07

on finite nonempty subsets, with singleton case $\square:B\to B$08. The axioms Fix and Ind force $\square:B\to B$09 to be the greatest fixed point of the monotone map

$\square:B\to B$10

so tangled closure is a greatest-fixed-point operator built from ordinary closure. Every complete closure algebra has a unique tangled expansion, but the completion theory becomes markedly worse: there exists a tangled closure algebra that embeds into no complete tangled closure algebra, hence has neither a MacNeille completion nor a spatial representation. In this precise sense, tangled closure algebras strictly generalize the McKinsey–Tarski closure-algebra framework (Goldblatt et al., 2016).

A second neighboring theory appears in rough-set duality. The relevant paper does not define MT-algebras by that name, but it studies monadic algebras $\square:B\to B$11, i.e. Boolean algebras with a normal additive operator satisfying

$\square:B\to B$12

where $\square:B\to B$13. These are the Boolean-algebraic counterparts of $\square:B\to B$14-equivalence frames and serve as the closest analogue of McKinsey–Tarski-style algebras in that setting. The duality is with approximation spaces $\square:B\to B$15, and the canonical relation on ultrafilters is given by

$\square:B\to B$16

This situates McKinsey–Tarski-style modal algebras within a Jónsson–Tarski/Kripke/van Benthem discrete-duality framework (Düntsch et al., 9 Jan 2026).

The classical algebraization of topology has also been criticized on model-theoretic grounds. Lipparini observes that closure-algebra homomorphisms do not coincide with continuous maps, since direct image generally satisfies only an inequality involving closure, not exact preservation. The proposed remedy is relational: encode topology by the preorder

$\square:B\to B$17

leading to specialization semilattices and specialization posets for which continuity is exactly homomorphism of the induced image maps (Lipparini, 2022). In a different direction, the broader Gödel–McKinsey–Tarski and Blok–Esakia program has been extended to intuitionistic strict implication by replacing the classical unary modal setting with Heyting–Lewis algebras and descriptive strict-implication frames (Groot et al., 2021). Together, these developments show that McKinsey–Tarski algebras occupy a central position in a larger constellation of algebraic, topological, modal, and categorical theories.