Pseudocomplementation in rings of continuous functions

PSEUDOCOMPLEMENT A TION IN RINGS OF CONTINUOUS FUNCTIONS GURAM BEZHANISHVILI AND MAR CUS TRESSL Abstra ct. W e study rings of real-v alued con tin uous functions in terms of pseudo- complemen tation conditions on v arious lattices attac hed to their prime sp ectrum. W e fully c haracterize pseudocomplementation in all cases and ha v e an almost complete c haracterization of relativ e pseudo complemen tation. Contents 1. In tro duction 1 2. Preliminaries 3 3. Pseudo complemen tation in bounded distributive lattices 6 4. F orests and ro ot systems 10 5. Pseudo complemen tation in rings of contin uous functions 12 References 18 1. Intr oduction The ring C ( T ) of contin uous real-v alued functions on a top ological space T is one of the most studied ob jects in top ology [ GiJe60 ]. It is a classic result of Gel’fand and K olmogorov [ GK91 , Theorem IV’] based on earlier work of Stone [ Sto37 ] that the Stone-Čec h compactification of a completely regular space X can b e describ ed as the maximal sp ectrum of C ( T ) and of its subring C ∗ ( T ) of b ounded functions. This has generated a comprehensive study of maximal sp ectra of the rings C ( T ) , as w ell as their minimal sp ectra, see for instance [ GiJe60 , HW04 ]. How ever, the complicated structure of full prime sp ectra Sp ec C ( T ) and Sp ec C ∗ ( T ) (as for instance in [ Sc h97 , T re07 ]) is in vestigated to a lesser degree. W e are contributing to this study in the following wa y . By Stone duality , prime sp ectra of rings are equiv alently describ ed by the lattice of their compact and op en subsets. W e aim to pro vide c haracterizations of v ariants of pseudo complemen tation prop erties of these lattices and their order duals in terms of algebraic prop erties of Date : March 31, 2026. 2020 Mathematics Subje ct Classific ation. Primary: 54C30, 06D15 Secondary: 54G05, 54G10, 18F70. Key wor ds and phr ases. Ring of (real-v alued) contin uous functions; basically disconnected space; P-space; metrizable space; pseudo complemented distributiv e lattice; Stone algebra; Heyting alge- bra; dualit y theory . 1 2 C ( T ) and topological prop erties of Sp ec C ( T ) . F or instance, we hav e complete c har- acterizations for pseudo complemen tation and an almost complete c haracterization of relativ e pseudocomplementation. In order to carry these out, w e recall and add to the knowledge of pseudocomplementation in lattice theory . This is done in sections 3 and 4 , which we describ e no w. The study of pseudo complemen tation and relative pseudo complemen tation has b een a mainstream topic in lattice theory [ BaDw74 , Grä11 ], which has imp ortan t consequences in algebraic logic as such structures serve as algebraic mo dels of v arious non-classical logics [ RS63 , Ras74 ]. In particular, there is a well-dev elop ed dualit y theory for such structures, which utilizes Stone dualit y [ Sto37 ] and Priestley duality [ Pri70 ] for b ounded distributiv e lattices, the tw o b eing t wo sides of the same coin [ Cor75 ]. F rom Ho c hster’s c haracterization of prime sp ectra of comm utative rings [ Hoc69 ], it follo ws that sp ectra of b oth b ounded distributiv e lattices and of commutativ e rings describ e the same class of top ological spaces, the so-called sp e ctr al sp ac es (see [ DST19 ] for a detailed accoun t). There are v arious characterizations of when a b ounded distributiv e lattice L is pseudo complemen ted or a Stone algebra using either the sp ectral space of L or its sister Priestley space (see [ Grä63 , ChGr69 , Pri74 , Pri75 ]), as w ell as when it is a Heyting algebra or a coHeyting algebra (see [ Esa74 , Esa19 , BBGK10 , DST19 ]). In this pap er we utilize the ab o ve results, as w ell as the structure of Sp ec C ( T ) , to obtain sev eral characterizations of when the lattice ˚ K (Sp ec C ( T )) of compact op en sets of Sp ec C ( T ) is pseudo complemented or a coHeyting algebra. Our main results include the following characterizations: Let T b e a completely regular space. Then • ˚ K (Sp ec C ( T )) is pseudo complemen ted iff X is basically disconnected iff ˚ K (Sp ec C ( T )) is a Stone algebra (Theorem 5.6 ). As a consequence we also sho w that ˚ K (Sp ec C ( T )) is pseudo complemen ted iff ˚ K (Sp ec C ∗ ( T )) is pseudo- complemen ted (Corollary 5.7 ) • If T is metrizable, then ˚ K (Sp ec C ( T )) is a Heyting algebra iff X is discrete (Corollary 5.9 ). • In Theorem 5.2 w e sho w that the order dual of ˚ K (Sp ec C ( T )) is pseudo com- plemen ted iff it is a Heyting algebra iff it is a Stone algebra iff X is a P-space. F urthermore, all these conditions on Sp ec C ( T ) are equiv alen t to their form u- lations for the subspace of z-prime ideals z - Sp ec C ( T ) (see 2.4 for definitions). On the w ay we will also see ho w kno wn results fit into the topic of pseudo complem- n tation. F or instance, on the ring theoretic side we reframe kno wn c haracterizations of basically disconnected spaces ( 5.6 ) and Baer rings ( 3.7 ); on the lattice theoretic side, Theorem 3.5 com bines and extends v arious known c haracterizations of Stone algebras (see 3.6 ). 3 2. Preliminaries In this preliminary section w e briefly recall Stone and Priestley dualities for b ounded distributiv e lattices. The former giv es rise to sp ectral spaces, while the latter giv es rise to Priestley spaces; we briefly discuss the Cornish isomorphism b et ween the tw o categories. W e also recall prime sp ectra of commutativ e rings. Our notation mostly follo ws that in [ DST19 ]; for the reader interested in the frame theoretic approach to the matter we refer to [ Joh86 ]. 2.1 . Sp ectral Spaces Let X b e an arbitrary topological space. F or x, y ∈ X w e sa y that x sp ecializes to y , and write x ⇝ y , if y ∈ { x } . It is well known (and easy to chec k) that the sp ecialization relation ⇝ is alw ays a preorder, and that it is a partial order iff X is a T 0 -space. F or S ⊆ X we call Sp ez( S ) = { x ∈ X | x ∈ { s } for some s ∈ S } the set of sp ecializations of S (in X ) and Gen( S ) = { x ∈ X | s ∈ { x } for some s ∈ S } the set of generalizations of S (in X ). Define X max = { x ∈ X | { x } = { x }} and X min = { x ∈ X | Gen( x ) = { x }} . If X is the prime sp ectrum of a ring, then X max is the set of maximal ideals of the ring. Since w e are primarily in terested in this example, w e read ⇝ as ≤ (instead of the alternativ e reading as ≥ ). Therefore, Sp ez( S ) = ↑ S and Gen( S ) = ↓ S . W e set O ( X ) = op en subsets of X, ˚ K ( X ) = { U ∈ O ( X ) | U is compact } , K ( X ) = { X \ U | U ∈ ˚ K ( X ) } , K ( X ) = the Bo olean algebra generated b y ˚ K ( X ) in the p ow erset of X . Recall that X is a sp ectral space if it is sob er and ˚ K ( X ) is a b ounded sublattice and a basis for O ( X ) . F or a spectral space X , w e write X inv for the inv erse space of X , whic h coincides with the de Gro ot dual of X (see, e.g., [ DST19 , 1.4.1] ). Therefore, ˚ K ( X inv ) = K ( X ) . F urthermore, w e write X con for the patc h space of X (see, e.g., [ DST19 , 1.3.11] ). Th us, X con is a Bo olean space (compact, Hausdorff, and zero- dimensional) and ˚ K ( X con ) = K ( X ) . Prop erties of these top ologies will b e referred to with the respective decoration, lik e ‘in versely compact’ (meaning ‘compact in X inv ’ ) or ‘patch closed’ (meaning ‘closed in X con ’). The Stone represen tation of b ounded distributive lattices states that eac h suc h lattice L is isomorphic to the lattice ˚ K ( X ) for some spectral space X , which is unique up to a homeomorphism. There are v arious w ays to see this (see, e.g., [ DST19 , sections 3.1–3.3]). W e c ho ose X to b e the set X = PrimI( L ) = { p ⊆ L | p is a prime ideal of L } , whose topology is given by the basis consisting of the sets D ( a ) := { p ∈ X | a / ∈ p } for a ∈ L ; w e write V ( a ) := { p ∈ X | a ∈ p } . Then X is a sp ectral space and the 4 map L − → ˚ K ( X ) , a 7→ D ( a ) is a lattice isomorphism. The definition of the space PrimI( L ) lo oks formally very similar to the definition of the prime sp ectrum of a ring; as the latter is our target, w e chose this presentation. By definition w e see that p ⇝ q ⇐ ⇒ p ⊆ q . Th us, X max is the space of maximal ideals of L . The represen tation then gives rise to lattice isomorphisms L inv − → ˚ K ( X inv ) = K ( X ) , A − → ˚ K ( X con ) = K ( X ) , where L inv is the order dual of L , and A is the Bo olean env elop e of L (see, e.g., [ DST19 , 3.4.2] ). The morphisms in the category Spec of sp ectral spaces are sp ectral maps , i.e. (con tinuous) maps f : X − → Y with the prop ert y V ∈ ˚ K ( Y ) ⇒ f − 1 ( V ) ∈ ˚ K ( X ) . Stone duality sa ys that Spec is an ti-equiv alen t (ak a dually equiv alen t) to the cate- gory of b ounded distributiv e lattices with bounded lattice morphisms. 2.2 . Priestley spaces A Priestley space is a Bo olean space X equipp ed with a partial order ≤ that satisfies the Priestley separation axiom : x ≤ y = ⇒ ∃ D clopen down-set : x / ∈ D and y ∈ D . The Priestley represen tation then states that eac h b ounded distributive lattice L is isomorphic to the lattice of clop en down-sets of some Priestley space, which is unique up to an order-homeomorphism. In particular, if X is the sp ectral space of L , then ( X con , ⇝ ) is the Priestley space of L , where X con is the patch space of X and ⇝ is the specialization order. The morphisms in the category Priestley of Priestley spaces are con tin uous and order preserving maps and Priestley dualit y sa ys that Priestley is anti-equiv alen t to the category of b ounded distributive lattices with b ounded lattice morphisms. As is evident, there is a close connection b et ween sp ectral spaces and Priestley spaces. Indeed, eac h sp ectral space X gives rise to the Priestley space ( X con , ⇝ ) , and the top ology of the sp ectral space is recov ered as the top ology of op en do wn- sets of ( X con , ⇝ ) . As w as demonstrated by Cornish in [ Cor75 ], this corresp ondence extends to an isomorphism of Spec and Priestley , see also [ BBGK10 ] and [ DST19 , 1.5.15] . Here is a dictionary connecting terminology from sp ectral spaces and Priestley spaces: Let X b e a spectral space and let P = ( X con , ≤ ) b e the corresponding Priestley space; recall that x ≤ y ⇐ ⇒ y ∈ { x } . (1) The patc h closed subsets of X (also called pr o c onstructible in [ DST19 ]) are the closed subsets of P . (2) The closure in X of a patch closed set S is ↑ S . (3) The closure in the in verse top ology of a patc h closed set S is ↓ S . (4) The elements of K ( X ) are called constructible sets and they are precisely the clopen subsets of P . (5) The op en subsets of X are the op en down-sets of P . If U ⊆ X , then U is compact open iff U is op en and constructible iff U is a clop en down-set of P . 5 (6) A set S is compact in X if and only if ↓ S is patc h closed. 2.3 . Prime sp ectra of comm utativ e rings W e adopt the notational conv en- tions of [ DST19 , section 12], which also p oints to general texts on the prime sp ectrum of a ring and whic h might b e used for additional elemen tary facts ab out these. Let A be a ring, whic h alw ays means commutativ e and unital in this pap er. Recall that the prime sp ectrum (or Zariski sp ectrum) of A is the sp ectral space Sp ec( A ) of prime ideals of A with the top ology having the sets D ( f ) := { p ∈ Sp ec( A ) | f / ∈ p } as a basis; w e write V ( f ) := { p ∈ Sp ec( A ) | f ∈ p } . Sp ecialization in Sp ec( A ) is in- clusion and in the setup of this paper this means p ≤ q ⇐ ⇒ p ⊆ q ( ⇐ ⇒ q ∈ { p } ). The closed sets of Sp ec( A ) are those of the form V ( S ) := { p ∈ Sp ec( A ) | S ⊆ p } for S ⊆ A . There is an antitone Galois connection b et ween subsets of A and subsets of Sp ec( A ) mapping S ⊆ A to V ( S ) and Z ⊆ Sp ec( A ) to ⋂ Z ; this defines a bijection b et ween radical ideals of A and closed subsets of Sp ec( A ) . 2.4 . Prime sp ectra of rings of contin uous functions Let T b e a completely regular space. W e write C ( T ) for the ring of contin uous functions T − → R . Note that C ( T ) is also a p oset where f ≤ g means ∀ t ∈ T : f ( t ) ≤ g ( t ) . W e adopt the terminology from [ GiJe60 ]. (i) An ideal I of C ( T ) is called z-radical or a z-ideal if f ∈ I and Z ( g ) ⊇ Z ( f ) implies g ∈ I . Here Z ( f ) = { x ∈ T | f ( x ) = 0 } is the zero set of f . The complemen ts are called cozero sets and they form the bounded sublattice Coz( T ) of the frame O ( T ) of op ens (use Z ( f ) ∩ Z ( g ) = Z ( f 2 + g 2 ) ). (ii) The prime ideal sp ectrum Sp ec C ( T ) of C ( T ) is a sp ectral ro ot system, see [ DST19 , 13.2.3] . (iii) T is called an F-space if Spec C ( T ) is stranded , i.e. Sp ec C ( T ) is also a forest (see [ GiJe60 , 14.25]) and a P-space if Sp ec C ( T ) is Bo olean, see section 4 for details. (iv) A completely regular space T is called basically disconnected if the closure of any cozero set is op en. Suc h a space is zero-dimensional (i.e. the clop en sets form a basis) b ecause when x ∈ U ∈ O ( T ) , then there is V ∈ Coz( T ) with x ∈ V and V ⊆ U . (v) The map ε : T − → Sp ec C ( T ) that sends x ∈ T to m x := { f | f ( x ) = 0 } is an em b edding of top ological spaces and its image is con tained in the subspace β T of maximal ideals of C ( T ) . The space β T is a compact Hausdorff space and is called the Stone–Čech Compactification of T (cf. [ DST19 , 8.4.15] ). (vi) The prime z-ideals of Sp ec C ( T ) form a sp ectral subspace of Sp ec C ( T ) , de- noted b y z - Sp ec C ( T ) and this is the Stone dual of the lattice Coz( T ) . Hence Coz( T ) ∼ = ˚ K (z - Sp ec C ( T )) and K (z - Sp ec C ( T )) is (isomorphic to) the order dual of Coz( T ) . Each maximal ideal and ev ery minimal prime ideal of C ( T ) is z-radical. The space z - Spec C ( T ) is the patch closure of ε ( T ) . All this is more or less spread out in [ GiJe60 ] and in [ Sc h97 ]; w e refer to [ T re06 , p. 145] for a summary with more details. 6 The spaces considered ab ov e are displa yed in the follo wing diagram of spaces and subspaces, where we consider ε as inclusion and write Z = z - Sp ec C ( T ) : T β T = X max Z = T con X = Sp ec C ( T ) X min Here T con denotes the closure of T for the patc h top ology . T aking preimages under the em b edding T − → Z induces a lattice isomorphism ˚ K ( Z ) − → Coz( T ) . 3. Pseudocomplement a tion in bounded distributive la ttices In this section w e fo cus on the lattice theoretic side of pseudocomplementation and v ariants, as well as their translation into top ology using Stone duality . A partic- ular emphasis is giv en to Heyting algebras and Stone algebras. The results will b e deplo y ed in section 5 to c haracterize pseudo complemen tation of v arious lattices attac hed to rings of contin uous functions. 3.1 . Pseudo complementation Let L = ( L, ∧ , ∨ , ⊥ , ⊤ ) b e a b ounded distributiv e lattice. W e recall that the pseudo complemen t of a ∈ L is the largest elemen t of the set { x ∈ L | a ∧ x = ⊥} . W e write a ∗ for the pseudo complement when it exists. If all elemen ts of L ha ve a pseudo complemen t, then L is called pseudo complemented . [1] The follo wing theorem pro vides a c haracterization of the duals of pseudo complemen ted lattices: 3.2. Theorem. L et L b e a b ounde d distributive lattic e and X its sp ac e of prime ide als. The fol lowing c onditions ar e e quivalent: (i) L is pseudo c omplemente d. (ii) F or every U ∈ ˚ K ( X ) , the closur e U = ↑ U is c onstructible. (iii) The fol lowing two c onditions hold. [2] (a) F or U ∈ ˚ K ( X ) the op en r e gularization in t( U ) b elongs to ˚ K ( X ) , and (b) X min is quasi-c omp act; in this c ase X min is even a p atch close d subset of X , se e [ DST19 , 4.4.16] . Pr o of. The equiv alence of (i) and (ii) is prov ed in [ Pri74 , Prop osition 1]. F or the equiv alence of (ii) and (iii) see [ DST19 , 8.3.9] . □ F ollowing [ BBI16 ], we call the spectral spaces satisfying the condition in the ab o ve theorem PC-spaces ; in [ DST19 , 8.3.1] they are called semi-Heyting . W e next recall that a pseudocomplemented lattice L is a Stone algebra according to [ GS57 ] pro vided a ∗ ∨ a ∗∗ = ⊤ for eac h a ∈ L . Let X be a PC-space. F or U ∈ ˚ K ( X ) w e ha ve U ∗∗ = int( U ) (see [ DST19 , 8.3.10] ), yielding the dual c haracterization giv en in 3.5 of Stone algebras (cf. [ GS57 , DL59 , Pri74 ]). [1] If the pseudo complemen tation map is named, the resulting structure is a p-algebr a (see, e.g., [ Bly05 , section 7.1]). [2] Observ e that neither of these conditions can b e dropped, see [ DST19 , 8.3.11] . 7 3.3. R emark. Condition (b) in 3.2 alone also has a meaning in terms of complemen- tation, whic h go es as follows. In any sp ectral space X the set X min is compact if and only if for all U ∈ ˚ K ( X ) there is some V ∈ ˚ K ( X ) suc h that X min = U min · ∪ V min . This follows from [ DST19 , 4.4.16] , whic h sa ys that X min is compact iff X and X inv induce the same top ology on X min . The prop ert y X min = U min · ∪ V min resem bles the prop ert y of V b eing a “generic complemen t” of U in X and it is equiv alent to sa ying that U ∩ V = ∅ and U ∪ V is dense in X . 3.4 . Normal sp ectral spaces and retraction maps Normal sp ectral spaces are discussed in detail in [ DST19 , 8.4] . A sp ectral space is normal (in the sense of top ology) if and only if the closure of every p oin t contains a unique closed point. An- other wa y of saying this is that the inclusion map X max  → X p ossesses a retraction r : X − → X max that preserv es sp ecialization. In such a space X , the map r : X − → X max that sends x ∈ X to the unique closed point in { x } is a contin uous, closed, and prop er retraction of the embedding X max  → X [ CC83 , Prop osition 3, p. 230]. [3] No w note that for a subset S of X w e hav e Gen(Sp ez( S )) = ↓↑ S = r − 1 ( r ( S )) . Hence if S is closed, then ↓ S = ↓↑ S is closed as w ell. F or an arbitrary sp ectral space X we no w c haracterize the existence of a contin uous retraction X − → X min of the inclusion map X min  → X ; notice that this condition is not equiv alent to sa ying that X inv is normal. 3.5. Theorem. L et L b e a b ounde d distributive lattic e and X its sp ac e of prime ide als. The fol lowing c onditions ar e e quivalent: (i) L is a Stone algebr a; r e c al l that L is isomorphic to ˚ K ( X ) . (ii) F or al l U ∈ ˚ K ( X ) the closur e U = ↑ U is op en. (iii) X min is p atch close d and for al l x, y , z ∈ X , if y , z ≤ x then ther e is u ∈ X with u ≤ y , z . In other wor ds, ↑ ( D ∩ E ) = ↑ D ∩ ↑ E for any two down-sets D , E . (iv) X min is p atch close d and X inv is normal, i.e. for e ach x ∈ X ther e is a unique y ∈ X min such that y ≤ x . (v) X inv is normal and the map X − → X , sending x ∈ X to the unique minimal p oint sp e cializing to x , is sp e ctr al. (vi) Ther e is a c ontinuous r etr action X − → X min of the inclusion map X min  → X . Pr o of. (iv) ⇔ (iii) : Both implications follow from the fact that X = ↑ ( X min ) . (iv) ⇔ (v) holds by [ DST19 , 8.4.14] applied to X inv . (v) ⇒ (vi) is clear. (vi) ⇒ (ii) . By assumption, there is a con tinuous map s : X − → X min whose restric- tion to X min is the iden tit y map. T ak e U ∈ ˚ K ( X ) . Since U is a down-set, we know that ↑ U = ↑ ( X min ∩ U ) . In order to sho w that ↑ U is op en, w e deplo y contin uity of s , so it is sufficien t to pro ve ↑ U = s − 1 ( X min ∩ U ) . [3] A ring whose sp ectrum is normal is called a Gel’fand ring (ak a a PM-ring ). 8 ⊆ . If U ∋ u ≤ x , then tak e y ∈ X min ∩ ↓ u . By con tinuit y of s this implies s ( y ) ≤ s ( x ) and then y = s ( x ) b ecause s is the iden tity on X min and X min has no prop er sp ecializations. ⊇ . If s ( x ) ∈ X min ∩ U , then tak e y ∈ X min with y ≤ x ; w e see again that y = s ( y ) = s ( x ) ∈ U , th us x ∈ ↑ U . Hence we kno w that (iv) ⇒ (iii) ⇒ (v) ⇒ (vi) ⇒ (ii) . But (ii) implies that L is pseudo- complemen ted, using 3.2 . Therefore, for the remaining implications we may assume that L is pseudo complemen ted (and X min is patc h closed). No w w e may refer to [ Pri74 ] whic h under this assumption has already pro ved the equiv alences of (i) , (ii) , (iii) , and (iv) ; see Propositions 2 and 3 in that pap er. □ 3.6 . Remark (i) Grätzer and Sc hmidt in [ GS57 ] resolve G. Birkhoff ’s problem no. 70 [ Bir48 , p. 149] by showing that a pseudo complemen ted distributive lattice is a Stone algebra if and only if all distinct minimal prime ideals are coprime. The equiv- alence of (i) and (iv) in 3.5 implies this result. (ii) The equiv alence of (i) and (iii) in 3.5 also dates bac k to [ DL59 ], prov iding an alternate solution of Birkhoff ’s problem (see the paragraph after Theorem 3 in their pap er). (iii) In [ ChZa97 , Prop osition 2.37] posets satisfying the first order condition in 3.5(iii) are called str ongly dir e cte d . This condition is also kno wn as the c on- fluenc e or Chur ch-R osser condition [ BdR V01 , Example 3.44] and pla ys an imp ortan t role in Kripke semantics for mo dal logic. 3.7 . Application If X is the prime sp ectrum of a reduced comm utativ e ring A and U ∈ ˚ K ( X ) , then U is of the form D ( f 1 ) ∪ . . . ∪ D ( f n ) for some n ∈ N and f i ∈ A . Th erefore, condition (ii) of 3.5 is equiv alent to saying that for each f ∈ A the closure of D ( f ) is open. No w one c hecks without difficulty that the closure of D ( f ) is V (Ann( f )) and clop en subsets of X are of the form V ( e · A ) for some idemp oten t elemen t e ∈ A . Since Ann( f ) and e · A are radical ideals in an y reduced ring [4] , condition (ii) of 3.5 is equiv alent to saying that for every f ∈ A there is an idemp oten t elemen t e ∈ A with Ann( f ) = e · A . Reduced rings with this prop erty are called Baer rings in [ Kis74 ], which w e adopt here (other auth ors call suc h rings we ak Baer ). Hence for the prime sp ectrum X of a reduced comm utative ring A , 3.5 implies that A is a Baer ring if and only if X min is a retract of X (repro ving [ Kis74 , Theorem 2]), if and only if ˚ K ( X ) is a Stone algebra. Theorem 3.5 can also be form ulated in the language of the in v erse topology (utilizing also Theorem 3.2 ). W e only record those statements that are used later on. 3.8. Corollary . F or a sp e ctr al sp ac e X , the fol lowing ar e e quivalent: (i) K ( X ) is a Stone algebr a. (ii) F or al l C ∈ K ( X ) the closur e ↓ C of C in the inverse top olo gy is clop en. [4] If h 2 ∈ e · A , then h 2 (1 − e ) 2 = 0 , so h (1 − e ) = 0 and h = eh ∈ e · A . 9 (iii) X is normal and K ( X ) is pseudo c omplemente d. (iv) X is normal and X max is p atch close d. 3.9 . Relative pseudo complemen tation Let L b e a bounded distributiv e lattice and a, b ∈ L . W e recall that the relative pseudo complement of a with resp ect to b is the largest element of the set { x ∈ L | a ∧ x ≤ b } , [ RS63 , BaDw74 ], denoted b y a → b ; another name is implication . If a → b exists for all a, b ∈ L , then L is called relatively pseudo complemen ted or a Heyting algebra . The dual spaces of Heyting algebras are called Esakia spaces (or Heyting sp ac es in [ DST19 , section 8.3]). They w ere first describ ed b y Esakia [ Esa74 ] and further studied by n umerous authors. The next theorem is w ell known (see [ Esa74 , Esa19 , BBGK10 , BGJ13 , DST19 ]). 3.10. Theorem. L et L b e a b ounde d distributive lattic e and X its dual sp e ctr al sp ac e. The fol lowing ar e e quivalent: (i) L is a Heyting algebr a. (ii) The closur e of any c onstructible subset of X is c onstructible; r e c al l that the closur e of any p atch close d set is the upset that it gener ates. (iii) Every close d and c onstructible subsp ac e of X is a PC-sp ac e. (iv) F or any S ⊆ X , the closur e of S for the inverse top olo gy is the p atch closur e of ↓ S . (Notic e that in any sp e ctr al sp ac e, the closur e of a subset is the down-set gener ate d by the p atch closur e of that set.) Pr o of. The equiv alence of (i) and (ii) is in [ Esa74 , Theorem 1]. The equiv alence of (ii) and (iv) is in [ Esa19 , Theorem 3.1.2], which is a translation of the 1985 Russian original. The equiv alence of (ii) and (iii) is straightforw ard. □ 3.11 . Summary The relationship b et ween Bo olean algebras, Heyting algebras, Stone algebras, and pseudo complemen ted lattices can b e summarized as shown in the diagram b elo w. Recall that X = PrimI( L ) , L ∼ = ˚ K ( X ) = { compact op en sets } and K ( X ) = { constructible sets } . Let Cl : P ( X ) − → P ( X ) b e the closure operator of the p ow erset of X given b y the top ology of X . Also recall that L is a Bo olean algebra if and only if ev ery compact open set is closed. Heyting algebra Cl( K ( X )) ⊆ K ( X ) Boolean algebra Cl | K ( X ) = id K ( X ) pseudo complemented Cl( ˚ K ( X )) ⊆ K ( X ) X min compact and U ∈ ˚ K ( X ) ⇒ int( U ) ∈ ˚ K ( X ) Stone lattice Cl( ˚ K ( X )) ⊆ ˚ K ( X ) 10 4. F orests and root systems T o characterize pseudo complemen tation in prime sp ectra of rings of contin uous func- tions, we will need to lo ok at sp ectral root systems, i.e. sp ectral spaces for whic h ev- ery patch closed subset is normal. In this section we collect some information ab out suc h spaces and consequences of section 3 for the pseudo complemen ted context. 4.1 . Ro ot systems and forests (i) A sp ectral space is a sp ectral ro ot system if the closure of every p oint is a sp ecialization chain. In older literature this is called c ompletely normal , but this conflicts with the standard use of “completely normal” in top ology , see [ DST19 , 8.5.3(iv)] and the fo otnote of [ DST19 , 1.6.15] . If w e read sp ecial- ization as a partial order ≤ , this means that the sp ecialization p oset defined b y X is a ro ot system. By [ DST19 , 8.5.1] a sp ectral space is a spectral ro ot system iff every patch closed subspace is a normal top ological space. (ii) Dually , X is a sp ectral forest if X inv is a sp ectral ro ot system. 4.2 . Stranded spaces A T 0 -space is stranded if its specialization p oset is a sum of chains in the category of p osets; in other words, if the sp ecialization relation is a forest and a ro ot system. In particular, stranded sp ectral spaces are sp ectral ro ot systems and sp ectral forests. 4.3 . Remark T rees, forests, and ro ot systems play an imp ortan t role in mo dal logic. (i) First completeness results in mo dal logic with resp ect to trees go back to [ DL59 ] and [ Kri63 ]. Extensions of the intuitionistic propositional calculus IPC complete with respect to forests were in vestigated b y Drugush (see, e.g., [ Dru82 , Dru84 ]). As is argued in [ V ar97 ], it is the tree mo del prop ert y that is resp onsible for decidability of mo del-c hec king in mo dal logic. F or further results in this direction, w e refer to [ ChZa97 ] and [ BdR V01 ]. (ii) The Heyting algebras whose dual sp ectra are ro ot systems are kno wn as Gö del algebr as (see, e.g., [ H ´ 98 , section 4.2]) The corresp onding logic was introduced in [ Dum59 ] under the name LC, and is now kno wn as the Gö del-Dummett lo gic . This is a prominent extension of IPC whic h has b een studied extensively (see, e.g., [ ChZa97 , H ´ 98 ]). Its bi-intuitionistic v ersion bi-LC is exactly the logic of stranded posets [ BMM24 ]. F ree algebras in the corresp onding v arieties of Gö del algebras hav e b een thoroughly inv estigated. W e reference [ BMM24 ] and [ Car26 ] for a detailed accoun t and relev an t references. 4.4. Prop osition. (i) If X is a sp e ctr al r o ot system, then the fol lowing c onditions ar e e quivalent. (a) X inv is an Esakia sp ac e. (b) U max is p atch close d for al l U ∈ ˚ K ( X ) . (c) F or al l U, V ∈ ˚ K ( X ) the set U max ∩ V is a c omp act subset of X . (ii) If X is a sp e ctr al for est, then X is Esakia if and only if C min is c omp act for al l C ∈ K ( X ) . 11 Pr o of. (i) Since X is a sp ectral ro ot system, each U ∈ ˚ K ( X ) is a normal sp ectral space. Hence X inv is Esakia ⇐ ⇒ eac h A ∈ K ( X inv ) is a PC-space, b y [ DST19 , 8.3.4] , ⇐ ⇒ for all U ∈ ˚ K ( X ) the space U inv is a PC-space , ⇐ ⇒ for all U ∈ ˚ K ( X ) the set U max is patc h closed , b y 3.8(iii) ⇔ (iv) using that U is normal , ⇐ ⇒ for all U, V ∈ ˚ K ( X ) the set U max ∩ V is compact, b y [ DST19 , 8.4.14] . (ii) is dual to (i), taking into accoun t that the set of minimal p oin ts of a sp ectral space is patch closed if and only if it is compact. □ 4.5. Observ ation. L et X b e a sp e ctr al sp ac e. (i) Supp ose that every maximal p oint of X is in the p atch closur e of the set of minimal p oints of X . Then the fol lowing c onditions on X ar e e quivalent: (a) X is Bo ole an, (b) ˚ K ( X ) is a Stone algebr a, (c) X is Esakia, (d) X is a PC-sp ac e, (e) X min is p atch close d. (ii) Supp ose that every minimal p oint of X is in the p atch closur e of the set of maximal p oints of X . [5] Then the fol lowing c onditions on X ar e e quivalent: (a) X is Bo ole an, (b) K ( X ) is a Stone algebr a, (c) X inv is Esakia, (d) X inv is a PC-sp ac e, (e) X max is p atch close d. Pr o of. Item (ii) is (i) spelled out for X inv . F or item (i), first note that the implications (a) ⇒ (b),(c) and (b) ⇒ (d), (c) ⇒ (d) are ob vious, and the implication (d) ⇒ (e) holds b y 3.2(iii) . The implication (e) ⇒ (a) follo ws from the assumption that all maximal p oin ts of X are in the patc h closure of X min , hence there are no sp ecializations in X under assumption (e), and so X is Bo olean. □ The follo wing consequence of our results s o far – display ed for sp ectral spaces – will giv e us in 5.2 full information ab out pseudo complementation properties for in verse sp ectra of rings of con tinuous functions: 4.6. Theorem. L et X b e a sp e ctr al r o ot system such that every minimal p oint of X is in the p atch closur e of the set of maximal p oints of X . The fol lowing c onditions ar e e quivalent. (i) X is Bo ole an. (ii) K ( X ) is a Stone algebr a. (iii) The inverse sp ac e of X is Esakia, henc e K ( X ) is a Heyting algebr a. (iv) The inverse sp ac e of X is a PC-sp ac e, henc e K ( X ) is pseudo c omplemente d. (v) X max is p atch close d. (vi) F or al l U ∈ ˚ K ( X ) the set U max is p atch close d. (vii) F or al l U, V ∈ ˚ K ( X ) the set U max ∩ V is c omp act. [5] Using [ DST19 , 4.4.6] , this condition is equiv alen t to saying that X max is dense in X . 12 (viii) F or every C ∈ K ( X ) the set ↓ C is op en (e quivalently: is p atch op en) in X . (ix) F or every C ∈ K ( X ) the set ↓ C is clop en in X . Pr o of. Conditions (i) – (v) are equiv alen t by 4.5 (ii). Conditions (iii) , (vi) and (vii) are equiv alent by 4.4 . Conditions (ii) , (iv) and (viii) are equiv alent by 3.2 applied to X inv . The implication (ix) ⇒ (viii) is a weak ening. (viii) ⇒ (ix) . Since X is a spectral ro ot system, it is normal, hence w e may apply 3.4 , whic h tells us that ↓ C = r − 1 ( C ) is closed. It is open b y (viii) , thus it is clop en. □ 5. Pseudocomplement a tion in rings of continuous functions In this final section w e pro ve our main results on rings of contin uous functions. W e start with a reminder on the structure of Sp ec C ( T ) . 5.1 . F act Let T b e a completely regular space. Recall from 2.4 that we write C ( T ) for the ring of con tinuous functions T − → R . (i) (See, e.g., [ Sc h97 , section 5]) The em b edding T  → β T induces an em b ed- ding ι : C ( β T )  → C ( T ) of rings (via restriction), its image is the ring C ∗ ( T ) of bounded con tinuous functions T − → R and Sp ec( ι ) : Sp ec C ( T ) − → Sp ec C ( β T ) is a homeomorphism onto an in versely closed (i.e., patc h closed and closed under generalizations) subset S of Sp ec C ( β T ) . This embedding extends the embedding T  → β T (when w e iden tify T with its image in Spec( C ( T )) as in 2.4(v) ) (and similarly for β T ). W e get a comm utative diagram of embed- dings of spaces: Sp ec C ( T ) Sp ec C ( β T ) T β T Spec( ι ) F urthermore, the map that sends a maximal ideal m of C ( T ) to the Jacobson radical of ι − 1 ( m ) is a homeomorphism b et w een the space of maximal ideals of C ( T ) and C ( β T ) (Gelfand-K olmogoroff theorem, a v ailable in a muc h broader con text, see [ T re07 , Thm. 10.1]); the embedding β T  → Sp ec C ( β T ) is a home- omorphism on to the space of maximal ideals. (ii) Finally , the space (Sp ec C ( β T )) \ S is stranded (but is in general not sp ectral) and is equal to the set of proper generalisations of the image of Sp ec( ι ) . More precisely , the maximal c hains in (Spec C ( β T )) \ S are all of the form { p ∈ Sp ec C ( β T ) | ι − 1 ( m ) ⊊ p } , where m runs through the maximal ideals of C ( T ) . This follo ws from [ T re07 , Thm. 10.5] and also from results in [ Sc h97 ]. Here is a picture of Spec C ( β T ) showing ho w Sp ec C ( T ) (iden tified with the image of Sp ec( ι ) , shown in the blue area) sits inside Sp ec C ( β T ) . Closed p oin ts (=maximal ideals=maximal p oin ts in the sp ectral space terminology) 13 sit on top, “specialization go es upw ards”; Sp ec C ( T ) and Sp ec C ( β T ) hav e homeomorphic minimal sp ectra. (Sp ec C ( β T )) \ Sp ec C ( T ) F or a completely regular space T , let Y b e one of the spaces Sp ec( C ( T )) , Sp ec( C ( T )) inv , z - Spec( C ( T )) , or z - Spec( C ( T )) inv . W e will characterize the four conditions on ˚ K ( Y ) in the diagram of 3.11 as w ell as the condition “ Y min is compact” by means of topological prop erties of T . W e start with Spec( C ( T )) inv and z - Sp ec( C ( T )) inv . These are c haracterized by T b eing a P-space and is deduced from 4.6 : 5.2. Theorem. L et T b e a c ompletely r e gular sp ac e, X = Sp ec C ( T ) and Z = z - Sp ec C ( T ) . The fol lowing c onditions ar e e quivalent. (i) T is a P-sp ac e, i.e. X is Bo ole an and ˚ K ( X ) is a Bo ole an algebr a. (ii) Z is Bo ole an, henc e K ( Z ) [6] is a Bo ole an algebr a. (iii) K ( Z ) is a Stone algebr a. (iv) The inverse sp ac e of Z is Esakia, henc e K ( Z ) is a Heyting algebr a. (v) The inverse sp ac e of Z is a PC-sp ac e, henc e K ( Z ) is pseudo c omplemente d. (vi) Z max is p atch close d. (vii) F or al l U ∈ ˚ K ( Z ) the set U max is p atch close d. (viii) F or al l U, V ∈ ˚ K ( Z ) the set U max ∩ V is c omp act. (ix) F or every C ∈ K ( Z ) the set ↓ C is op en (e quivalently: is p atch op en) in Z . (x) F or every C ∈ K ( Z ) the set ↓ C is clop en in Z . F urthermor e these c onditions ar e e quivalent to every c ondition (ii) – (x) when they ar e formulate d for X inste ad of Z . Pr o of. W e ha ve X min ⊆ X max con = Z b y 2.4(vi) , in particular Z max = X max and its patc h closure con tains Z min = X min . Therefore, by 4.6 , condition (ii) is equiv alent to each of the conditions (iii) – (x) . Similarly , condition (i) is equiv alent to each of the conditions (iii) – (x) formulated for X instead of Z . Hence it remains to sho w that (i) is equiv alent to (ii) . But this follows from X min , X max ⊆ Z and the fact that a sp ectral space is Bo olean precisely when all p oin ts are minimal and maximal. □ [6] Recall from 2.4(vi) that K (z - Sp ec C ( T )) is the order dual of Coz( T ) . 14 W e no w fo cus on Spec( C ( T )) and z - Spec( C ( T )) . 5.3 . Cozero complemen tation A completely regular space is called cozero complemen ted if for ev ery cozero set U of T there is a cozero set V of T with U ∩ V = ∅ such that U ∪ V is dense in T . Since Coz( T ) ∼ = ˚ K (z - Sp ec C ( T )) and T is patc h dense in z - Sp ec C ( T ) (see 2.4(vi) ), this is equiv alen t to saying that for every op en and quasi-compact subset of z - Sp ec C ( T ) there is some op en and quasi-compact subset V of z - Sp ec C ( T ) with U ∩ V = ∅ such that U ∪ V is dense in z - Sp ec C ( T ) . Hence b y 3.3 , T is cozero complemented if and only if (z - Sp ec C ( T )) min is quasi- compact. The terminology and the original pro of of the equiv alence here is from [ HJ65 ] and v arious further equiv alent conditions may b e found in [ HW04 , Theorem 1.3]. F or cozero complementation in more general con texts w e refer to [ BDM11 , KLMS09 , MZ03 , ST10 ]. W e see from 3.2 that cozero complementation of T is a weak ening of Sp ec C ( T ) b eing a PC-space, whic h is obtained from dropping condition (iii)(a) there. 5.4 . Pseudo complementation of cozero sets Let Z = z - Sp ec C ( T ) . Recall that ˚ K (z - Sp ec C ( T )) ∼ = Coz( T ) , and hence all pseudo complemen tation conditions considered here are on Coz( T ) . Now Coz( T ) is pseudo complemen ted pro vided for eac h cozero set U of T there is a largest cozero set V of T with U ∩ V = ∅ . Using complete regularity of T , this is the same as saying that T \ U itself is a cozero set. Hence, z - Sp ec( C ( T )) is a PC-space ⇐ ⇒ ∀ U ∈ Coz( T ) : U is a zero set . Suc h spaces are called we ak Oz-sp ac es in [ Aul84 ], see also [ BDGWW09 ]. F or instance, Oz-sp ac es in the sense of [ Bla76 ] (like perfectly normal spaces, i.e., those spaces for which Coz( T ) = O ( T ) ) and metric spaces are w eak Oz spaces. In order to characterize when Sp ec C ( T ) is a PC-space, we require the follo wing: 5.5. Prop osition. L et T b e a c ompletely r e gular sp ac e and let X ⊆ Sp ec C ( T ) b e p atch close d and c onvex for sp e cialization, i.e. p ⊆ q ⊆ r and p , r ∈ X imply q ∈ X . The fol lowing ar e e quivalent: (i) X is a PC-sp ac e. (ii) ˚ K ( X ) is a Stone algebr a. (iii) X min is p atch close d and X is str ande d. Pr o of. (iii) ⇔ (ii) follo ws from 3.5 because the sp ectral ro ot system X is stranded iff X inv is normal. The implication (ii) ⇒ (i) holds by definition. (i) ⇒ (iii) Assume that X is a PC-space. By 3.2 w e know that X min is patc h closed. Supp ose X is not stranded. Since X is a sp ectral ro ot system, there must b e p oin ts p  = q in X min and a common specialization of p and q in X . Because X is con v ex, w e know 1 / ∈ p + q . By [ T re06 , Prop osition 3.9] (and [ GiJe60 , 14B]), the sum p + q of the ideals p and q is a z-prime ideal. As X is conv ex, we get p + q ∈ X . Since X min is Hausdorff, there is some U ∈ ˚ K ( X ) with q ∈ U and p / ∈ U . Since p is minimal in X , w e get p / ∈ Spez( U ) ∩ X = U ∩ X . Let C = { p } ∩ U ∩ Gen( p + q ) . 15 Since Sp ec C ( T ) is a sp ectral ro ot system, C is a chain, whic h is not empty b ecause p + q ∈ C . Because X is con vex, and p , p + q ∈ X , w e kno w C ⊆ X . No w, the nonempty patch closed c hain C has a smallest element, whic h w e denote b y r . As r ∈ U there is some q 0 ∈ U ∩ X min with q 0 ⊆ r . As p / ∈ U , the prime ideals p and q 0 are incomparable. By [ T re06 , Prop osition 3.9] again, p + q 0 is a z-radical prime ideal of C ( T ) . Therefore, p + q 0 ∈ Gen( r ) ∩ C and the minimality of r in C implies that r = p + q 0 is a z-radical prime ideal with r ∈ X . Let S =  { p } ∩ Gen( r )  \ { r } . Here is the depiction of the situation, where the lines represen t inclusion, the blue p oin ts are in U and the green p oin ts are in C . p q 0 q r = p + q 0 p + q S C Since p  = r , we get p ∈ S ⊆ { p } , hence S is a nonempty c hain. Since p , r ∈ X and X is con vex, w e also see that S ⊆ X . The minimality of r in C implies S ∩ U = ∅ , hence S ⊆ X \ U . Since X is a PC-space, the set U ∩ X is constructible in X , so X \ U is patc h closed. Because r ∈ X ∩ U , we obtain r / ∈ S con . W e now sho w that r ∈ S con , whic h giv es the desired contradiction. Since S is a chain, it suffices to sho w that S has no largest element with resp ect to inclusion b ecause then the c hain S has the suprem um r and this has to be in S con . In order to see that S has no largest elemen t, let I ∈ Sp ec( C ( T )) and let f ∈ C ( T ) with I ⊆ r and f ∈ r \ I . Then I ⊊ J = p I + f · C ( T ) ∈ Sp ec( C ( T )) , but J  = r by [ T re07 , Lemma 14.1] since r is a z-ideal. □ 5.6. Theorem. L et T b e a c ompletely r e gular sp ac e. The fol lowing ar e e quivalent: (i) Sp ec( C ( T )) is a PC-sp ac e. (ii) ˚ K (Sp ec C ( T )) is a Stone algebr a. (iii) Sp ec( C ( T )) min is p atch close d and T is an F -sp ac e, i.e. Sp ec( C ( T )) is str ande d. [7] (iv) T is b asic al ly disc onne cte d. (v) The p oset C ( T ) is De dekind σ -c omplete. (vi) F or every f > 0 in C ( T ) , the interval [0 , f ] is pseudo c omplemente d. Remark. The equiv alence of the last four conditions is not new and is recorded here to sho wcase the top ological impact of the first tw o conditions. Pr o of. (i) ⇔ (ii) ⇔ (iii) holds b y 5.5 applied to X = Sp ec C ( T ) . The equiv alence (iii) ⇔ (iv) essen tially is the equiv alence of the conditions (iv) and (i) in 3.5 applied [7] See [ GiJe60 , Theorem in 14.25] for more information on F-spaces. 16 to the sp ectral root system z - Sp ec( C ( T )) b ecause the definition of basically discon- nected says that Coz( T ) is a Stone algebra and z - Sp ec( C ( T )) and Sp ec( C ( T )) hav e the same minimal sp ectrum and one of them is stranded iff the other one is. Ho w- ev er, our assertion (iii) ⇔ (iv) here is not new: Recall from [ HJ65 , Theorem 5.3(e)] that for any F -space T , Sp ec( C ( T )) min is patch closed if and only if T is basically disconnected. [8] This shows (iii) ⇒ (iv) . Con versely , b y [ GiJe60 , 14N, p. 215], every basically disconnected space is an F -space, hence (iv) also implies (iii) . The equiv alence of (iv) with eac h of (v) and (vi) is not new and b elongs to the theory of Ab elian  -groups: In [ LZ71 , pp. 283–287] it is shown that a completely regular space T is basically disconnected (named ‘principal pro jection prop erty’ in that source) if and only if C ( T ) , seen as a p oset, is Dedekind σ -complete, if and only if C ( T ) , viewed as a lattice ordered group, is pro jectable , i.e. for ev ery f > 0 in C ( T ) the in terv al [0 , f ] is pseudocomplemented. F or a reference and some explanation of the terminology , see [ W yn07 , Proposition 2.1]. □ 5.7. Corollary . Spec C ( T ) is a PC-sp ac e if and only if Sp ec C ( β T ) is a PC-sp ac e. Pr o of. The spaces Sp ec C ( T ) and Sp ec C ( β T ) hav e homeomorphic minimal sp ectra, see 5.1(ii) . By [ GiJe60 , Theorem in 14.25], T is an F -space if and only if β T is an F -space. Hence, the desired equiv alence follo ws from 5.6 . Alternativ ely one can use [ GiJe60 , 6M, p. 96], whic h says that T is basically disconnected if and only if β T is basically disconnected and then deploy 5.6 . □ 5.8. Corollary . L et T b e a c ompletely r e gular sp ac e. If Sp ec C ( T ) is a PC-sp ac e, then no se quenc e ( x n ) n ∈ N of distinct p oints of T has a limit in T . In p articular, e ach metrizable subsp ac e of T is discr ete. Pr o of. By [ GiJe60 , 14N.1], no point of an F-space is the limit of a sequence of distinct points. Now apply 5.6 . □ As a consequence of 5.8 w e obtain: 5.9. Corollary . If T is a metric sp ac e, then Sp ec C ( T ) is a PC-sp ac e if and only if it is an Esakia sp ac e if and only if T is discr ete. 5.10. Corollary . L et T b e a c ompletely r e gular sp ac e. Then T is a P -sp ac e if and only if T is b asic al ly disc onne cte d and ↓ V ( f ) ∩ (Sp ec C ( T )) min is op en in (Sp ec C ( T )) min for every f ∈ C ( T ) . Pr o of. If T is a P -space, then T is basically disconnected (see 2.4 ) and ↓ V ( f ) = V ( f ) is clop en. F or the conv erse, we w ork in X = Sp ec C ( T ) and use 4.6 to show that ↓ V ( f ) is constructible for all f ∈ C ( T ) : By 5.6 we know that T is an F -space and X min is patch closed. It follo ws that the restriction r 0 : X min − → X max of the natural retraction r : X − → X max is a bijectiv e contin uous map b et w een compact spaces. Hence r 0 is a homeomorphism. [8] Observ e that any basically disconnected space is cozero complemented: If U is a cozero set, then U is open, hence is clopen and so is the zero set of its c haracteristic function. In comparison: Sp ec( C ( T )) is Boolean if and only if all cozero sets are closed, cf. [ GiJe60 , 14.29]. 17 Since ↓ V ( f ) is closed, the assumption implies that ↓ V ( f ) ∩ X min is clop en in X min . Consequen tly , the set V ( f ) ∩ X max = r 0 ( ↓ V ( f ) ∩ X min ) is clop en in X max . Since r is a sp ectral map X − → X max (see [ DST19 , 8.4.13] and use that X max is Bo olean), this sho ws that ↓ V ( f ) = r − 1 ( V ( f ) ∩ X max ) is constructible as w ell. □ 5.11 . Pseudo complementation conditions for some sp ecial spaces Let S b e the space exhibited in [ HJ65 , Example 5.8], where it is shown that S is cozero complemen ted but Coz( S ) is not pseudo complemented. By [ HJ65 , Example 5.8], the space β N \ N do es not ha v e compact minimal spectrum, in fact no p oin t of (Sp ec C ( β N \ N )) min has a compact neighborho o d. Since z - Sp ec C ( β N \ N ) is nat- urally homeomorphic to a closed and constructible subset of z - Sp ec C ( β N ) [9] , 3.10 implies that the latter is not an Esakia space. Using this information we present a table with an o v erview of the pseudo com- plemen tation conditions w e ha ve considered. As ab o ve, X = Sp ec( C ( T )) for a completely regular space T and Z = z - Sp ec( C ( T )) . The conditions in ro ws with an asterisk in the second column are equiv alent for all c hoices of T (not only those sho wn in the last four columns). Similarly for the conditions in rows with a dagger. The n um b ers b ehind an entry give a reference that implies the answer. Note that if X is PC, then also Z is PC as follows from [ DST19 , 8.3.19] , whic h implies that Z is ev en a PC-subspace of X . ⇔ R n β N β N \ N S from 5 . 11 X Stone ∗ N, 5.6 Y, 5.6 N, 5.6 N, 5.6 Z Stone, i.e. T basically disconnected ∗ N, 5.6 Y, 5.6 N, 5.6 N, 5.6 X Esakia N, 5.6 ? N, 5.11 N, 5.11 Z Esakia Y, 5.4 N, 5.11 N, 5.11 N, 5.11 X PC ∗ N, 5.8 Y, 5.7 N, 5.6 N, 5.6 Z PC Y, 5.4 Y, 5.7 N, 5.6 N, 5.6 X min compact † Y, 5.3 Y, 5.1 N, 5.11 Y, 5.11 Z min compact † Y, 5.3 Y, 5.1 N, 5.11 Y, 5.11 Also recall from 5.2 that for the inv erse spaces of X and Z all conditions are equiv- alen t to T b eing a P-space. Op en problem. W e do not ha ve a characterization of those completely regular spaces T for whic h Sp ec( C ( T )) is an Esakia space. In 5.9 w e ha ve seen that metric spaces hav e this prop ert y only if they are discrete, but we ha ve no example of an infinite compact Hausdorff space T suc h that Sp ec( C ( T )) is an Esakia space. In particular, w e do not know whether Sp ec( C ( β N )) is Esakia. W e also do not ha ve a general characterization of when z - Sp ec( C ( T )) is Esakia (equiv alently: Coz( T ) is a Heyting algebra). This does happ en frequently , e.g. when [9] By the Tietze extension theorem, the restriction map C ( β N ) − → C ( β N \ N ) is surjective and one verifies without difficulty that its kernel is the z -ideal I generated by the unique extension of the function 1 n from N to β N ; hence z - Sp ec C ( β N \ N ) ∼ = V ( I ) ∩ z - Sp ec C ( β N ) . 18 all op en sets of T are cozero sets, lik e in p erfectly normal spaces, but we do not hav e a full c haracterization. References [Aul84] C. E. Aull. Embeddings extending v arious types of disjoin t sets. Ro cky Mountain J. Math., 14(2):319–330, 1984. 14 [BaDw74] Ra ymond Balb es and Philip Dwinger. Distributive lattices. 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