Conjunctive Join Semi-Lattices
CConjunctive Join-Semilattices
Delzell, Charles N. [email protected]
Madden, James J. [email protected]
Ighedo, Oghenetega [email protected]
June 9, 2020
One of the most general extensions of Stone’s Representation Theorem for BooleanAlgebras concerns distributive join-semilattices, i.e., join-semilattices in which anyfinite cover of any element has a refinement that exactly covers that element. Dis-tributivity thus defined assures that a join-semilattice has sufficiently many primeideals to represent it as the join-semilattice of all compact opens in some sober T -space in which every open is a union of compact opens. See [G], Chapter II, section5. The present paper includes a representation theorem for join-semilattices thatneed not be distributive but instead (or in addition) are conjunctive . A join-semilattice L is said to be conjunctive (or have the conjunction property) if it has a top element1 and it satisfies the following first-order condition: for any two distinct a, b ∈ L ,there is c ∈ L such that either a ∨ c (cid:54) = 1 = b ∨ c or a ∨ c = 1 (cid:54) = b ∨ c . Equivalently,one can show, a join-semilattice is conjunctive if every principal ideal is an inter-section of maximal ideals. We present simple examples showing that a conjunctivejoin-semilattice may fail to have any prime ideals. We show that every conjunctivejoin-semilattice is isomorphic to a join-closed subbase for a compact T -topology onmax L , the set of maximal ideals of L . The representation is canonical, in that whenapplied to a join-closed subbase for a compact T -space X , the space produced bythe representation is homeomorphic with X .For any ideal I ⊆ L there is a join-semilattice morphism φ : L → M and amaximal ideal J of M such that I = φ − ( J ). We say a join-semilattice morphism1 a r X i v : . [ m a t h . L O ] J un : L → M is conjunctive if φ − ( w ) is an intersection of maximal ideals of L whenever w is a maximal ideal of M . We show that every conjunctive morphism betweenconjunctive join-semilattices is induced by a multi-valued function from max M tomax L . Thus, the representation is functorial in a manner that is reminiscent ofdistributive lattices or rings, but with some unexpected and intriguing modifications.The conjunction property—or more precisely its dual, the “disjunction property”—first appeared in Wallman’s 1938 paper [W] as a condition on a distributive latticethat guarantees the existence of sufficiently many maximal filters to distinguish be-tween elements of the lattice. This property was generalized by Pierce in his 1954paper [P], in a way that is meaningful in any semigroup. Simmons and Macnab[S] (1978) seems to have been the first to use of the term “conjunctive” with thesame meaning as in the present paper, but they applied it to distributive lattices,not to join-semilattices in general. Johnstone mentions their work and uses theirterminology in [J84b]. The conjunction property is closely related to the propertyof subfitness, introduced in 1973 by Isbell [I] as a separation axiom for locales. Sub-fitness is discuused at length in Chapter V of the book [PP], but here again, thoughthe definition is the same, it is applied only to frames (i.e., complete lattices in which ∧ distributes over all suprema).A conjunctive join-semilattice must have a top element, but applications (suchas those discussed in [MZ]) require more generality. We define a new property, idealconjunctivity , that enables us to extend the representation theory to join semilatticesthat do not have 1. Ideal conjunctivity is a generalization to join-semilattices of thenotion of “finite subfitness” that was introduced in [MZ]; see subsection 3.4, below.Every join semilattice L with 1 has a smallest congruence (called R ( L )) forwhich the quotient is conjunctive, and R ( L ) is the maximal congruence in whichthe congruence class of 1 is a singleton. (This is easily deduced from results in [P].)This situation carries over to distributive lattices and to compact frames, since R ( L )respects meets and infinite joins in these settings. The study of this congruence in thecontext of point-free topology was initiated by Johnstone in [J84], where he used it tomodify the lattice of ideals of a bounded distributive lattice L to produce the locale of“almost maximal ideals” of L . As is commonly done in point-free topology, Johnstoneused a nucleus to represent the congruence. (Recently, Haykazyan [H] has studiedthis further.) Banaschewski and Harting [B ] considered the congruence for generalcompact frames, calling the quotient mapping the “saturation,” and Banaschewskicontinued the study of the saturation in [B ]. A manifestation of Pierce’s theoremin frames is stated in Corollaries V.1.3.3 and V.1.4.1 of [PP].Here is an outline of the present paper. We have included some expository ma-terial, especially in subsections 2.3 and 4.1, to provide continuity between our work2nd background literature and to fill in a few details that are not addressed in othersources.Section 2 begins with a review of basic definitions. An ideal of a join-semilatticeis a ∨ -closed downset; a prime ideal is a proper ideal whose complement is a filter.We provide an example of a finite join-semilattice with three maximal ideals andno prime ideals. After this, we review facts about the complete lattice of all idealsof a join-semilattice. In general, it may fail to satisfy any distributive laws, but itmay be characterized as the solution to a universal mapping problem. In the thirdsubsection, we present Pierce’s theory [P] specialized to join-semilattices. Finally,we introduce and discuss the new concept of ideal conjunctivity , which generalizesthe concept of conjunctivity to join-semilattices without top element.In Section 3, we prove the Representation Theorem for Conjunctive Join Semi-lattices and its partial generalization to ideally conjunctive join semilattices. Thenwe apply the theory to prove a far-reaching generalization of a result of Martinezand Zenk concerning “Yosida frames.”In Section 4, we review the representation theory for distributive join-semilattices,then show that a complete conjunctive join-semilattice is distributive if and only ifall its maximal ideals are prime. It is an open question as to whether this is truewithout the completeness hypothesis.The final section considers two problems that relate to distributive lattices. Abase for a topological space is said to be annular if it is a lattice. A Wallman basefor a space X is an annular base such that for any point u in any basic open U ,there a basic open V that misses u and together with U covers X . It is easy toshow that every Wallman base is conjunctive. We give an example of a conjunctiveannular base that is not Wallman. Finally, we examine the free distributive lattice dL over a conjunctive join semilattice L . In general, it is not conjunctive, but weshow that dL/R ( dL ) is isomorphic to wL := the sub-lattice of the topology ofthe representation space that is generated by L . The passage from L to wL is notfunctorial.The present paper demonstrates that the category of conjunctive join-semilatticesand conjunctive morphisms has a rich and interesting theory and that much of theexisting theory of conjunctive distributive lattices and subfit frames springs fromthe properties of these more-elementary structures. The foundational role of join-semilattices in the theory of frames and locales was highlighted in [JT], where theauthors suggested the following fruitful analogy:frames : rings : : complete join-semilattices : abelain groups.Few authors, however, have built on this. We hope that our work will inspire others3o explore the role of conjunctive join-semilattices in pointfree topology and otherareas. We have included a number of unanswered questions that appear at the endsof Sections 1 through 4. A join-semilattice is a set L equipped with an associative, commutative, idempotentbinary operation ∨ . L is partially ordered by the relation x ≤ y , which by definitionmeans x ∨ y = y . In this order, x ∨ y the least upper bound of x and y . Thelargest (respectively, smallest) element of L , if it contains one, is denoted by 1 or1 L (respectively, 0 or 0 L ). We say U ⊂ L is an up-set if a ∈ U and a ≤ b implies b ∈ U . The up-set { b ∈ L | a ≤ b } is denoted by ↑ a . Down-sets and ↓ a are definedanalogously. Definition 2.1.1.
Let L be a join-semilattice.( i ) We call a subset I ⊂ L a join-semilattice-ideal (or simply an ideal when thecontext is clear) if it is a down-set and a ∨ b ∈ I whenever a, b ∈ I .( ii ) We call an up-set a filter if it is non-empty and contains a lower bound for anytwo of its elements. We say an ideal I ⊆ L is prime if it is non-empty and itscomplement in L is a filter.( iii ) If a ∈ L , then an ideal of L is said to be a -maximal if it does not contain a ,and any properly larger ideal does contain a . When L has a top element 1,a 1-maximal ideal is called simply maximal . When L lacks a top element, aproper ideal that is contained in no larger proper ideal is said to be maximalproper . Lemma 2.1.2.
Suppose L is a join-semilattice and a ∈ L . Every proper ideal of L that does not contain a is contained in an a -maximal ideal.Proof. Suppose C is an increasing chain of ideals of L , none of which contains a .Evidently (cid:83) C does not contain a , and it is a downset closed under ∨ , so it is anideal. This shows that every increasing chain of ideals not containing a is containedin an ideal not containing a . The lemma therefore follows from Zorn’s Lemma.4 xample 2.1.3. A maximal ideal of a join-semilattice need not be prime.
Let X be the set { x, y, z } , and let L = { xyz, xy, xz, yz, ∅} ⊆ P X , where we abbreviate thesubset { x, y } as xy . The maximal ideals of L are m z := { xy, ∅} , m y := { xz, ∅} ,and m x := { yz, ∅} . Observe that L \ m z = { xyz, xz, yz } , and note that xz and yz ,though not in m z , have no lower bound in L \ m z . Thus, m z is not prime. Similarly,neither are the other maximal ideals. The only proper ideal of L other than the threemaximal ideals is {∅} , which is also not prime since its complement is not a filter;the improper ideals L and ∅ are of course not prime. Thus, while every nonemptyjoin-semilattice must have at least one maximal ideal (by Lemma 2.1.2), it need nothave any prime ideals..Let := { , } be the join semilattice with 0 ∨ L and any ideal I ⊆ L , define φ I : L → by setting φ I ( a ) = 0 if a ∈ I and φ I ( a ) = 1if a (cid:54)∈ I . Then φ I preserves ∨ by the definition of ideal. In contrast, if L is a latticeand I is proper, then φ I is a lattice morphism (i.e., preserves both ∨ and ∧ ) if andonly if I is prime. Id L Throughout the remainder of this paper, L denotes a join-semilattice. Sometimesadditional conditions are imposed. The set of all ideals of L , including the improperideals ∅ and L , is denoted by Id L . Since any intersection of ideals is an ideal, Id L isa complete lattice (with meet being set-theoretic intersection and join being the meetof all upper bounds). For any subset X ⊂ L , the intersection of all ideals containing X is denoted by (cid:104) X (cid:105) . Note that (cid:104){ x }(cid:105) = ↓ x . Evidently, (cid:104) X (cid:105) = { y ∈ L | y ≤ (cid:87) X (cid:48) for some finite X (cid:48) ⊆ X } . Observe that Id L , though it is a lattice, does not generally satisfy any distributivelaws. The equational law:for all a ∈ Id L and all B ⊆ Id L , a ∧ (cid:95) B = (cid:95) { a ∧ b | b ∈ B } holds if and only if L satisfies the distributive axiom for join-semilattices —see Sec-tion 4.The injection map ↓ : L → Id L is a join-semilattice morphism since ↓ a ∨ ↓ b = (cid:104)↓ a ∪ ↓ b (cid:105) = ↓ ( a ∨ b ). To simplify notation, we sometimes identify L with its imagein Id L . For example, if I ∈ Id L and a ∈ L , I ∨ a is understood to mean (cid:104) I ∪ { a }(cid:105) .5 emma 2.2.1. Let J be a complete join-semilattice. If f : L → J is a ∨ -morphism,then there is a unique morphism f : Id L → J that preserves all suprema and satisfies f ◦ ↓ = f .Proof. Define f : Id L → J by f ( I ) := (cid:87) { f ( y ) | y ∈ I } . If A ⊆ Id L , then f ( (cid:87) A ) = (cid:87) { f ( y ) | y ≤ (cid:87) A, A a finite subset of (cid:83)
A } = (cid:87) { f ( y ) | y ∈ I, I ∈ A } , since f preserves finite ∨ s , = (cid:87) { f ( I ) | I ∈ A } For any a ∈ L , f ( ↓ a ) = (cid:87) { f ( y ) | y ≤ a } = f ( a ). Thus f ◦ ↓ = f .Recall that an element c of a complete join semilattice is said to be compact if:whenever c ≤ (cid:87) X for some subset X of the lattice, there is a finite subset X (cid:48) of X such that c ≤ (cid:87) X (cid:48) . A join-semilattice is said to be algebraic if it is complete (andhence has 0 and 1) and is generated by its compact elements. The following factsare well-known; see [B], VIII.5.( i ) Let K be a complete join-semilattice. The set of compact elements, denoted bycpt K , forms a sub-join-semilattice of K containing 0. For any element a ∈ K ,let C ( a ) := { c ∈ cpt K | c ≤ a } . Then C ( a ) is an ideal of cpt K , and themap C : K → Id cpt K is order-preserving. For any ideal I ⊆ cpt K , we have (cid:87) I ∈ K since K is complete. The map (cid:87) : Id cpt K → K is order-preserving.( ii ) For any join semi-lattice L , I ∈ Id L is compact if and only if I = ∅ or I = ↓ a for some a ∈ L . Since every ideal is the supremum of the principal ideals in it,Id L is algebraic.( iii ) If A is an algebraic join-semilattice, then (cid:87) C ( a ) = a for all a ∈ A and C ( (cid:87) I ) = I for all I ∈ cpt A . Hence, the maps C and (cid:87) are inverses ofone another, giving an order-isomorphism A ∼ = Id cpt A . In this subsection, we present results of [P], specialized to join semilattices. Weinclude proofs because they do not take up much space, and it is useful to have themat hand in a notation that is consistent with the rest of this paper.We say R ⊆ L × L is a ∨ -congruence on L if R is an equivalence relation andfor all a, a (cid:48) , b ∈ L , ( a, a (cid:48) ) ∈ R implies ( a ∨ b, a (cid:48) ∨ b ) ∈ R . The set of equivalenceclasses of R is denoted by L/R , and the class of a ∈ L is denoted by [ a ], or [ a ] R if6eference to R is needed. The rule [ a ] ∨ [ b ] := [ a ∨ b ] defines an operation on L/R ,since if ( a, a (cid:48) ) , ( b, b (cid:48) ) ∈ R , then a ∨ b ∼ a (cid:48) ∨ b ∼ a (cid:48) ∨ b (cid:48) . With this operation, L/R isa join-semilattice, and a (cid:55)→ [ a ] : L → L/R is a surjective semilattice morphism. If
R, R (cid:48) are ∨ -congruences on L , we say R is weaker than R (cid:48) or R (cid:48) is stronger than R if R ⊆ R (cid:48) . The weakest ∨ -congruence on L is equality and the strongest is L × L .Let Y be a subset of L . We define the relation R Y = R Y ( L ) by:( a, a (cid:48) ) ∈ R Y ⇔ def ∀ x ∈ L, x ∨ a ∈ Y ⇔ x ∨ a (cid:48) ∈ Y. We define ( Y : a ) := { x ∈ L | x ∨ a ∈ Y } . We call the elements of ( Y : a ) the Y -supercomplements of a . If 1 ∈ L , we call theelements of (1 : a ) := ( { } : a ) simply the supercomplements of a . Observe that( a, a (cid:48) ) ∈ R Y if and only if ( Y : a ) = ( Y : a (cid:48) ). Lemma 2.3.1.
For any subset Y of L , R Y is a ∨ -congruence on L .Proof. R Y is clearly an equivalence relation. Suppose ( a, a (cid:48) ) ∈ R Y ( L ) and b ∈ L .Then for all x ∈ L , x ∨ ( b ∨ a ) = ( x ∨ b ) ∨ a ∈ Y ⇔ ( x ∨ b ) ∨ a (cid:48) = x ∨ ( b ∨ a (cid:48) ) ∈ Y .Thus ( a ∨ b, a (cid:48) ∨ b ) ∈ R Y , as required. Definition 2.3.2.
A congruence of the form R Y will be called a Pierce congruence (in recognition of [P]). We write R b as shorthand for R ↑ b . We say L is conjunctive if L has 1 and R ( L ) is equality.There are several equivalent ways to formulate the conjunctivity condition. Specif-ically, the following are clearly equivalent:1. L is conjunctive.2. If two elements of L have the same supercomplements, they are equal.3. If a and a are distinct elements of L , then there is w ∈ L such that either w ∨ a = 1 (cid:54) = w ∨ a , or w ∨ a (cid:54) = 1 = w ∨ a .
4. For all a, b ∈ L such that b (cid:54)≤ a , there is w ∈ L such that a ≤ w < w ∨ b .5. For all a, b ∈ L such that a < b , there is w ∈ L such that a ≤ w < w ∨ b .7. Every principal ideal of L is an intersection of maximal ideals. (See Proposition3.1.1.) Example 2.3.3.
A product of conjunctive join-semilattices is conjunctive. A sub-semilattice of a conjunctive join-semilattice need not be conjunctive. The two-element join-semilattice := { , } is conjunctive, as is × , but the sub-join-semilattice { (0 , , (1 , , (1 , } ⊆ × is not conjunctive. Remark.
It follows from the definitions that for any a ∈ L and any up-set U ⊆ L ,( U : a ) is an up-set of L . The map a (cid:55)→ ( U : a ) is order-preserving, if we orderup-sets by containment. Thus, a (cid:55)→ ( U : a ) is an order-isomorphism of L/R U onto { ( U : a ) | a ∈ L } . Note that ( U : a ) ∪ ( U : b ) is not generally equal to( U : a ∨ b ) = (( U : a ) : b ) = (( U : b ) : a ). For example, if L is the power set of { x, y } ,(1 : { x } ) ∪ (1 : { y } ) does not contain ∅ , but (1 : { x, y } ) = (1 : 1) does. Theorem 2.3.4.
Suppose U ⊆ L is an up-set. Then: ( i ) U is an R U class (so L/R U = U ). ( ii ) L/R U is conjunctive. ( iii ) In any congruence properly stronger than R U , the top class properly contains U .Proof. ( i ) If a ∈ U , ( U : a ) = L . If b (cid:54)∈ U , b ∨ b (cid:54)∈ U , so ( U : b ) (cid:54) = L .( ii ) We use [ a ] to denote [ a ] R U . Suppose [ a ] (cid:54) = [ b ]. Interchanging a and b ifnecessary, we may assume that there is c such that a ∨ c (cid:54)∈ U and b ∨ c ∈ U . Then[ a ] ∨ [ c ] (cid:54) = U and [ b ] ∨ [ c ] = U . Thus, [ a ] and [ b ] are in different classes of R ( L/R U ).( iii ) If R (cid:48) is properly stronger than R U , then ( a, a ∨ b ) ∈ R (cid:48) \ R U for some a, b ∈ L .Therefore, there is c ∈ L such that a ∨ c (cid:54)∈ U but a ∨ b ∨ c ∈ U . But a ∨ c ≡ a ∨ b ∨ c mod R (cid:48) , so the R (cid:48) equivalence class containing U also contains an element not in U . Lemma 2.3.5.
Suppose K is a conjunctive join-semilattice and h : L → K is asurjective ∨ -morphism. Let V = h − (1 K ) . Then [ x ] R V (cid:55)→ h ( x ) : L/R V → K is anisomorphism.Proof. Note that V := h − (1 K ) is an up-set. Let [ a ] denote [ a ] R V . For all a, a (cid:48) ∈ L ,the following are equivalent: ( i ) [ a ] = [ a (cid:48) ]; ( ii ) for all x ∈ L , x ∨ a ∈ V ⇔ x ∨ a (cid:48) ∈ V ;( iii ) for all x ∈ L , h ( x ) ∨ h ( a ) = 1 K ⇔ h ( x ) ∨ h ( a (cid:48) ) = 1 K . Since K is conjunctiveand h is surjective, ( iii ) is equivalent to h ( a ) = h ( a (cid:48) ). Thus, the map [ a ] (cid:55)→ h ( a ) iswell-defined and injective. It is surjective and respects ∨ by hypothesis, so it is anisomorphism. 8 heorem 2.3.6. Suppose U ⊆ L is an up-set. Let Q be any congruence on L inwhich U is a class. Then, [ x ] Q (cid:55)→ [ x ] R U is a surjective ∨ -morphism from L/Q to L/R U . Thus R U is the strongest congruence on L in which U is a class.Proof. For convenience, set J := L/Q and let f : L → J be the canonical surjection.Let K := J/R , and let g : J → K be the canonical surjection. Let h = g ◦ f .By 2.3.4, ( i ), h − (1 K ) = U . By Lemma 2.3.5, g ( f ( x )) (cid:55)→ [ x ] R U : K → L/R U is anisomorphism, so f ( x ) = [ x ] Q (cid:55)→ [ x ] R U : J → L/R U is a surjective ∨ -morphism.The following proposition shows that the results above imply analogous resultsfor distributive lattices, if we strengthen the hypotheses on U . Proposition 2.3.7.
Suppose L is a distributive lattice, and U is a filter. Then R U is a lattice-congruence.Proof. We need to verify that if ( a, a (cid:48) ) ∈ R U , then ( a ∧ b, a (cid:48) ∧ b ) ∈ R U . This is seenas follows. For all x ∈ L : x ∨ ( a ∧ b ) ∈ U ⇔ ( x ∨ a ) ∧ ( x ∨ b ) ∈ U ⇔ ( x ∨ a ) ∈ U & ( x ∨ b ) ∈ U, since U is an up-set, ⇔ ( x ∨ a (cid:48) ) ∈ U & ( x ∨ b ) ∈ U, since ( a, a (cid:48) ) ∈ R U , ⇔ ( x ∨ a (cid:48) ) ∧ ( x ∨ b ) ∈ U, since U is a filter, ⇔ x ∨ ( a (cid:48) ∧ b ) ∈ U. Proposition 2.3.8. If c ∈ L is a compact element, then R c preserves infinite joins.Proof. Suppose A , A ⊆ L and { [ a ] | a ∈ A } = { [ a ] | a ∈ A } . Then, for all x ∈ L : x ∨ (cid:95) A ≥ c ⇔ x ∨ (cid:95) A (cid:48) ≥ c, for some finite A (cid:48) ⊆ A ⇔ x ∨ (cid:95) A (cid:48) ≥ c, for some finite A (cid:48) ⊆ A ⇔ x ∨ (cid:95) A ≥ c. Recall that L refers to an arbitrary join-semilattice. Since Id L is a join-semilattice,all the results of the previous subsection concerning Pierce congruences apply to it.This must be understood with care. First of all, the superscript 1 in R (Id L ) refersto 1 Id L = L ∈ Id L . Second, while Id L is complete, the congruence R (Id L ) in9eneral respects only finite suprema. The canonical map a → [ a ] : Id L → (Id L ) /R preserves finite suprema, but it may fail to preserve infinite suprema, since 1 Id L isnot compact when L does not have a top element. Example 2.4.1.
View N as a join semilattice with the natural order. Then Id N = { ↓ n | n ∈ N }∪{ N } , where N = 1 Id N . For every n ∈ N , ( N : ↓ n ) = { N } . On the otherhand, ( N : N ) = Id N . Thus, Id N /R ∼ = . We have (cid:87) { ↓ n | n ∈ N } = N . However,[ ↓ n ] = [ ↓
0] for all n ∈ N . Thus, it is not the case that (cid:87) { [ ↓ n ] | n ∈ N } = [ N ]. Themap I (cid:55)→ [ I ] : Id N → Id N /R does not preserve infinite suprema. Definition 2.4.2.
The restriction of R (Id L ) to L is denoted by R (Id L ) | L := R (Id L ) ∩ ( L × L ) . We say that L is ideally conjunctive if R (Id L ) | L is equality. Definition 2.4.3.
We say that W ∈ Id L is an ideal supercomplement of a ∈ L if W ∨ a = L , i.e., the ideal generated by W ∪ { a } is the improper ideal L .As with the conjunctive property, there are several ways to say that a join-semilattice is ideally conjunctive. The following are clearly equivalent:1. L is ideally conjunctive.2. If two elements of L have the same set of ideal supercomplements, they areequal.3. for any a , a ∈ L , if a (cid:54) = a , there is an ideal W ∈ Id L such that either W ∨ a = L (cid:54) = W ∨ a , or W ∨ a (cid:54) = L = W ∨ a .
4. For all a, b ∈ L such that b (cid:54)≤ a , there is W ∈ Id L such that a ∈ W (cid:54) = L = W ∨ b .5. For all a, b ∈ L such that a < b , there is W ∈ Id L such that a ∈ W (cid:54) = L = W ∨ b .6. Every principal ideal of L is an intersection of maximal proper ideals. (SeeProposition 3.3.1.) Lemma 2.4.4. If L has , then L is ideally conjunctive if and only if L is conjunctive.Proof. Suppose a, b ∈ L , b (cid:54)≤ a : ( ⇒ ) By hypothesis, there is a proper ideal W suchthat a ∈ W and W ∨ b = L . Therefore, there is u ∈ W such that u ∨ b = 1. Letting w = u ∨ a , we have a ≤ w < w ∨ b . ( ⇐ ). By hypothesis, there is w ∈ L suchthat a ≤ w < w ∨ b . Let W = ↓ w. Proposition 4 of [J84b] is a version of this lemma for distributive lattices.10 .5 Problems
1. It is natural to ask if L/ ( R (Id L ) | L ) is always ideally conjunctive. We can-not mimic the proof of Theorem 2.3.4, because Id( L/ ( R (Id L ) | L )) is not thesame as (Id L ) /R . We conjecture that L/ ( R (Id L ) | L ) may fail to be ideallyconjunctive.2. The definition of ideally conjunctive refers to ideals, so it is not a first-ordercondition in the language of join-semilattices (as is the definition of conjunc-tivity). However, this does not preclude the possibility that the property ofbeing ideally conjunctive is first-order. We conjecture that it is not. The purpose of the present section is to show that every conjunctive join-semilatticeis a join-semilattice of open sets forming a subbase for a compact T space, and thatany join-semilattice map between conjunctive join-semilattices that satisfies a certaintechnical condition induces a continuous relation between the representation spacesfrom which we can recover the map. Also, we show that every ideally conjunctivejoin-semilattice is a join-semilattice of open sets forming a subbase for a T space(not necessarily compact). As an application, we give a generalization of a theoremof Martinez and Zenk. Example 3.0.1.
A conjunctive join-semilattice may have maximal ideals that arenot prime.
Indeed, the join-semilattice L in Example 2.1.3 is conjunctive, as we cansee by writing out the supercomplements of each element:If a = 1 xy xz yz ∅ (1 : a ) L = L { , xz, yz } { , xy, yz } { , xy, xz } { } The representation theory for conjunctive join-semilattices rests on the followingproposition.
Proposition 3.1.1.
Suppose L is a ∨ -semilattice with . Then L is conjunctive ifand only if: for all a, b ∈ L such that b (cid:54)≤ a , there is a maximal ideal of L thatcontains a and does not contain b . roof. ( ⇒ ) Suppose a, b ∈ L and b (cid:54)≤ a . Select w ∈ L such that a ≤ w < w ∨ b = 1. There is a maximal ideal m that contains w (and hence a ). Because w ∨ b = 1 and w ∈ m , we have b (cid:54)∈ m . ( ⇐ ) Again, suppose a, b ∈ L and b (cid:54)≤ a . Let m be a maximal ideal such that a ∈ m and b (cid:54)∈ m . Then, there is u ∈ m such that u ∨ b = 1. Since a ∈ m , u ∨ a <
1. For w , take u ∨ a . For the remainder of this subsection, we assume L is a conjunctive join-semilatticethat contains at least two elements (including ). Let max L denote the set of allmaximal ideals of L . For each a ∈ L , let (cid:98) a : max L → { , } be defined by (cid:98) a ( m ) := (cid:40) , if a ∈ m ;1 , if a / ∈ m. In terms of the map φ defined after Example 2.1.3, we have (cid:98) a ( m ) := φ m ( a ). Lemma 3.1.2.
The map a (cid:55)→ (cid:98) a : L → { , } max L is an injective - ∨ -morphism.Proof. By Proposition 3.1.1, for any two different elements of L , then there is amaximal ideal of L that contains one and not the other, so the map is injective. It isclear that (cid:98) L . Let m ∈ max L . Then (cid:91) a ∨ b ( m ) = 0iff a ∨ b ∈ m iff a ∈ m & b ∈ m iff (cid:98) a ( m ) = 0 & (cid:98) b ( m ) = 0 iff (cid:0) (cid:98) a ∨ (cid:98) b (cid:1) ( m ) = 0. Thus, (cid:91) a ∨ b = (cid:98) a ∨ (cid:98) b . Definition 3.1.3.
For each a ∈ L , let coz a := { m ∈ max L | (cid:98) a ( m ) = 1 } = { m ∈ max L | a (cid:54)∈ m } . We call coz a the cozero set of a . Remark.
Throughout this paper, we use coz a to refer to a set of maximal ideals.Observe that (cid:98) a is the characteristic function of coz a . The notation spec a , whichoccurs in Theorem 4.1.4, has a similar definition, but it is a set of prime ideals.Since a join semilattice may have maximal ideals that are not prime and prime idealsthat are not maximal, in general there is no relationship. Under the distributivehypothesis (see Section 4), every maximal ideal is prime.Let W L be the weakest topology on max L in which coz a is open for each a ∈ L .Let Spec Max L denote max L with the topology W L . Lemma 3.1.4.
Spec
Max L is T and a (cid:55)→ coz a is an isomorphism of L with a subbasefor Spec
Max L that is closed under finite joins. roof. W L is T , because given any two maximal ideals, each fails to contain atleast one element of the other, so each is in a cozero set not containing the other.By definition of the topology, the cozero sets form a subbase. The map a (cid:55)→ coz a isan injective ∨ -morphism by the previous lemma. Remark.
Let L be a subset of P X . We say that L is a T subbase if for any x, y ∈ X ,there is a ∈ L such that x ∈ a and y (cid:54)∈ a . It is clearly the case that if L is a T subbase, then the topology that it generates is T . Conversely, if L is not a T subbase, then there are points x, y ∈ X such that for all a ∈ L , x ∈ a ⇔ y ∈ a .Then the same thing is true for finite intersections of elements of L , and all unionsof such sets—hence for the topology generated by L . Definition 3.1.5.
For any B ⊆ L , we let (cid:104) B (cid:105) denote the ideal generated by B , andlet (cid:104)(cid:104) B (cid:105)(cid:105) denote the intersection of all maximal ideals containing B . Example 3.1.6.
In general, (cid:104) B (cid:105) may be a proper subset of (cid:104)(cid:104) B (cid:105)(cid:105) . For example,suppose L is the usual topology of [0 , L are the points of [0 , ,
1] that omit a neighborhood of 0 is an ideal thatis properly contained in m = the maximal ideal generated by (0 , Lemma 3.1.7.
The following are equivalent: ( i ) coz a ⊆ (cid:83) { coz b | b ∈ B } ;( ii ) a ∈ (cid:104)(cid:104) B (cid:105)(cid:105) ;( iii ) ∀ x ∈ L : x ∨ a = 1 ⇒ ∃ b ∈ (cid:104) B (cid:105) such that x ∨ b = 1 . Proof. ( i ⇔ ii ) Observe that (cid:91) { coz b | b ∈ B } = { m ∈ max L | B (cid:54)⊆ m } = { m ∈ max L | (cid:104)(cid:104) B (cid:105)(cid:105) (cid:54)⊆ m } . Thus, ( i ) ⇐⇒ (cid:0) ∀ m ∈ max L, a (cid:54)∈ m ⇒ (cid:104)(cid:104) B (cid:105)(cid:105) (cid:54)⊆ m (cid:1) ⇐⇒ ( ii ).( iii ⇒ ii ) Let m be a maximal ideal containing B . Toward a contradiction, sup-pose a (cid:54)∈ m . Then x ∨ a = 1 for some x ∈ m . Assuming ( iii ), it follows that thereis b ∈ (cid:104) B (cid:105) such that x ∨ b = 1. But this is impossible, since x and b are both in m .Hence, a ∈ m .( ii ⇒ iii ) Suppose a does not satisfy ( iii ). Then there is c ∈ L such that c ∨ a = 1,while B ∪ { c } is contained in a proper ideal, and hence is contained in a maximalideal m . Clearly, a (cid:54)∈ m , so a (cid:54)∈ (cid:104)(cid:104) B (cid:105)(cid:105) . 13 emma 3.1.8. Spec
Max L is compact.Proof. Setting a = 1 L in the previous lemma and letting x be any element of B we see that coz 1 L ⊆ (cid:83) { coz b | b ∈ B } if and only if 1 ∈ (cid:104) B (cid:105) . Thus any cover ofSpec Max L (= coz 1 L ) by elements of the subbase { coz b | b ∈ L } has a finite subcover.Compactness of Spec Max L then follows from the Alexander subbase theorem. Lemma 3.1.9.
Suppose X is a set with at least two elements and L ⊆ P X is a T subbase for a compact topology on X . Suppose further that L contains X and isclosed under joins. Then L is conjunctive, and the map x (cid:55)→ m x := { a ∈ L | x (cid:54)∈ a } is a homeomorphism of X with Spec
Max L .Remark. The conclusions of the theorem are also true if X contains a single pointand L = P X . Proof.
First, we show that for each x ∈ X , m x is a maximal ideal of L . By the T hypothesis, m x is not empty. Suppose a ∈ L \ m x (so x ∈ a ). Since L is a T subbase,for every y ∈ X \ { x } , m x contains an open neighborhood b y of y . Because X iscompact, there is a finite set Y ⊂ X \ { x } such that X ⊆ a ∨ b , where b = (cid:87) { b y | y ∈ Y } ∈ m x . Second, we show that L is conjunctive. Suppose a, b ∈ L and b (cid:54)⊆ a . Pick x ∈ b \ a . Then m x contains a and does not contain b . By Proposition 3.1.1, L isconjunctive. Third, we show that the map x (cid:55)→ m x : X → Spec
Max L is bijective. Itis injective by the T subbase hypothesis. It is surjective, for suppose m ∈ Spec
Max L .Since m is an ideal, { X \ a | a ∈ m } is a family of closed subsets of X with the finiteintersection property, and since X is compact, there is at least one point that all thesesets have in common. There can be no more than one, since m is maximal. Finally, x (cid:55)→ m x is a homeomorphism: by definition, coz a = { m x | x (cid:54)∈ a } , so a (cid:55)→ coz a is abijection between subbases for the topologies on X and Spec Max L .The following theorem summarizes all the lemmas in this subsection. Theorem 3.1.10 (Representation Theorem for Conjunctive Join-Semilattices) . ( i ) Suppose L is a conjunctive join-semilattice with at least two elements. Let max L be the set of maximal ideals of L and let coz : L → P (max L ) a (cid:55)→ coz a := { m ∈ max L | a / ∈ m } . Then coz is a join-semilattice injection and its image coz L is a subbase for a compact T topology (which we call W L ) on max L . ii ) Suppose X is a non-empty set and T is a compact T topology on X . Further,suppose that L is a subbase for T that is closed under finite unions. Then L isconjunctive and x (cid:55)→ m x := { a ∈ L | x / ∈ a } is a homeomorphism of ( X, T ) with (max L, W L ) . We begin by summarizing the functorial nature of Stone’s representation for dis-tributive lattices. Let A be a bounded distributive lattice. Spec A denotes the set ofprime ideals of A . For each a ∈ A , define (cid:98) a : Spec A → { , } by (cid:98) a ( p ) = (cid:40) a ∈ p φ : A → B is a morphism of bounded distributive lattices. For q ∈ Spec B ,define f φ ( q ) := φ − ( q ) ∈ Spec A . Then f φ is a function from Spec B to Spec A .Moreover (cid:98) a ◦ f φ = (cid:100) φ ( a ). The usual topology on Spec A is the weakest topology inwhich (cid:98) a − (1) is open for all a ∈ A (and similarly for B ). With respect to thesetopologies, f φ is continuous, for f − φ ( (cid:98) a − (1)) = ( (cid:98) a ◦ f φ ) − (1) = (cid:100) φ ( a ) − (1), so theinverse image of any basic open is open.If we attempt to replicate this in the category of conjunctive join semilattices(using maximal ideals rather than prime ideals), some modifications are necessary.First, there is no hope of representing every morphism of conjunctive join-semilatticesfor the following reason. We have mentioned above that for any ideal I in a join-semilattice L , the map φ I : L → { , } defined by φ I ( a ) = 0 iff a ∈ I is a 1- ∨ -morphism, and we have given examples that illustrate that there is a conjunctivejoin-semilattice L with an ideal I ⊆ L that is not an intersection of maximal idealsof L . For such an ideal, there is evidently no way to determine φ I ( a ) for each a ∈ L from (cid:98) a , since the only information we can extract from (cid:98) a is the set of maximal idealsto which a belongs. We address this problem by excluding such deviant morphismsfrom consideration. We will seek representations only for morphisms that satisfy thefollowing definition. Definition 3.2.1.
Suppose φ : L → M is a 1- ∨ -morphism of join-semilattices. Wesay that φ is a conjunctive morphism if φ − ( w ) is an intersection of maximal idealsof L whenever w is a maximal ideal of M .The second problem is that even when φ is conjunctive, taking inverse imagesdoes not yield a function from Spec Max M to Spec Max L . We must deal with the fact15hat for w ∈ Spec
Max M , φ − ( w ) is in general only an intersection of maximal ideals.We address this problem by replacing f φ with a multi-valued function Q φ . Using arelation Q φ in place of a function f φ raises a third problem. How do we compose (cid:98) a with a relation? The idea is to use the join in the complete semilattice , as we showin the next paragraph.To simplify notation, we use X L as shorthand for Spec Max L . Let X L and X M bethe representation spaces for conjunctive join-semilattices L and M . Suppose a ∈ L and (cid:98) a : X L → = { , } is its representation. Let Q ⊂ X M × X L be any relationsuch that for all w ∈ X M , there exists at least one v ∈ X L such that ( w, v ) ∈ Q .Thus, Q : w (cid:55)→ v is a “multi-valued function” from X M to X L . For any a ∈ L , define (cid:87) (cid:98) a ◦ Q : X M → by: (cid:95) (cid:98) a ◦ Q ( w ) := (cid:95) { (cid:98) a ( v ) | v ∈ X L & ( w, v ) ∈ Q } . Note that (cid:95) (cid:99) L ◦ Q = (cid:99) M . Now suppose b ∈ L and (cid:98) b : X L → is its representation. For any w ∈ X M , (cid:95) (cid:91) a ∨ b ◦ Q ( w ) = (cid:95)(cid:8) (cid:91) a ∨ b ( v ) (cid:12)(cid:12) v ∈ X L & ( w, v ) ∈ Q (cid:9) = (cid:95)(cid:8) (cid:98) a ( v ) ∨ (cid:98) b ( v ) (cid:12)(cid:12) v ∈ X L & ( w, v ) ∈ Q (cid:9) = (cid:95)(cid:8) (cid:98) a ( v ) (cid:12)(cid:12) v ∈ X L & ( w, v ) ∈ Q (cid:9) ∨ (cid:95)(cid:8) (cid:98) b ( v ) (cid:12)(cid:12) v ∈ X L & ( w, v ) ∈ Q (cid:9) = (cid:0) (cid:95) (cid:98) a ◦ Q (cid:1) ( w ) ∨ (cid:0) (cid:95) (cid:98) b ◦ Q (cid:1) ( w )Thus, we see that a (cid:55)→ (cid:87) (cid:98) a ◦ Q is a 1- ∨ -morphism from L to X M . Whether or not (cid:87) (cid:98) a ◦ Q lies in (cid:99) M depends, of course, on Q .We now show that for any conjunctive morphism φ : L → M between conjunctivejoin-semilattices, we have a natural relation Q φ that induces φ . Let Q φ := { ( w, v ) ∈ X M × X L | φ − ( w ) ⊆ v } . Proposition 3.2.2. If φ : L → M is a conjunctive morphism between conjunctivejoin-semilattices, then (cid:87) (cid:98) a ◦ Q φ = (cid:100) φ ( a ) for all a ∈ L .Proof. For any maximal ideal w of M , (cid:95) (cid:98) a ◦ Q φ ( w ) = (cid:95) { (cid:98) a ( v ) | v ∈ X L & φ − ( w ) ⊆ v } ,
16y the definitions of (cid:87) (cid:98) a ◦ Q and Q φ . Thus, for any w , (cid:95) (cid:98) a ◦ Q φ ( w ) = 0 ⇐⇒ for all v ⊇ φ − ( w ), (cid:98) a ( v ) = 0 ⇐⇒ for all v ⊇ φ − ( w ), a ∈ v ⇐⇒ a ∈ φ − ( w ) since φ is conjunctive ⇐⇒ φ ( a ) ∈ w ⇐⇒ (cid:100) φ ( a )( w ) = 0 . It is interesting to ask if Q − φ ( V ) := { w ∈ X M | ∃ v ∈ V φ − ( w ) ⊆ v } is an opensubset of Spec Max M , whenever V is an open subset of Spec Max L . Note first that w ∈ Q − φ (coz a ) ⇐⇒ ∃ v ∈ X L such that φ − ( w ) ⊆ v & a (cid:54)∈ v ⇐⇒ a (cid:54)∈ φ − ( w ) ⇐⇒ φ ( a ) (cid:54)∈ w ⇐⇒ w ∈ coz φ ( a ) . From this, we see that for any a ∈ L , Q − (coz a ) = coz φ ( a ). Second, note that for anycollection { A j } j ∈ J of subsets of X L , and any relation Q : X M → X L , Q − ( (cid:83) j ∈ J A j ) = (cid:83) j ∈ J Q − ( A j ). Intersections of cozero sets, however, are problematic. If a, b ∈ L : w ∈ Q − φ (coz a ∩ coz b ) ⇐⇒ ∃ v ∈ X L such that φ − ( w ) ⊆ v & a (cid:54)∈ v & b (cid:54)∈ v = ⇒ a (cid:54)∈ φ − ( w ) & b (cid:54)∈ φ − ( w ) ⇐⇒ φ ( a ) (cid:54)∈ w & φ ( b ) (cid:54)∈ w ⇐⇒ w ∈ coz φ ( a ) ∩ coz φ ( b ) . This shows that Q − (coz a ∩ coz b ) ⊆ coz φ ( a ) ∩ coz φ ( b ), but we see no reason toexpect equality, since the complement of φ − ( w ) need not be a filter. We do notknow if Q − (coz a ∩ coz b ) is open. Perhaps there is some other form of continuityfor multifunctions that Q satisfies. Example 3.2.3.
Let L be the distibutive lattice of all cofinite subsets of N , togetherwith 0 L = ∅ , so X L = { m x | x ∈ N } , where m x is the maximal ideal consisting ofall elements of L that do not contain x . Note that { L } is a prime ideal of L that isnot maximal, but it is the intersection of all the maximal ideals. Let M = { M , M } ,so X M is the one-point space, {∗} , where ∗ := { M } . Let φ : L → M satisfy φ ( a ) = 0 iff a = ∅ . Then, φ is a lattice morphism, and evidently Q φ = X M × X L .If a ∈ L , a (cid:54) = 0 L then Q − (coz a ) = {∗} . On the other hand, coz 0 L = ∅ and Q − (coz 0 L ) = ∅ = coz 0 M . 17 .3 Representing Ideally Conjunctive Join-semilattices We prove an analog of Proposition 3.1.1 for ideally conjunctive join-semilattices.
Proposition 3.3.1.
Suppose L is a join-semilattice. Then L is ideally conjunctiveif and only if: for all a, b ∈ L such that b (cid:54)≤ a , there is a maximal proper ideal of L that contains a and does not contain b .Proof. ( ⇒ ) Suppose a, b ∈ L and b (cid:54)≤ a . Select W ∈ Id L such that W ∨ b = 1 and a ∈ W (cid:54) = L . Clearly, b (cid:54)∈ W , so there is an ideal W (cid:48) containing W and maximalmissing b . Such a W (cid:48) is maximal proper, because any ideal that properly contains W (cid:48) contains both W and b and hence is equal to L . ( ⇐ ) Suppose a and b are differentelements of L . Without loss of generality, b (cid:54)≤ a . Let m be a maximal ideal of L thatcontains a and does not contain b . Then m is a supercomplement of b but not of a ,so a and b have different sets of supercomplements.The following corollary is analogous to part ( i ) of the Representation Theoremfor Conjunctive Join-Semilattices (Theorem 3.1.10). This corollary is weaker in thatit does not assert that (max L, W L ) is compact. The proof parallels that of Theorem3.1.10. Corollary 3.3.2.
Suppose L is an ideally conjunctive join-semilattice with at leasttwo elements. Let max L be the set of maximal proper ideals of L and let coz : L → P (max L ) a (cid:55)→ coz a := { m ∈ max L | a / ∈ m } . Then coz is a join-semilattice injection and its image coz L is a subbase for a T topology W L on max L .Remark. If L is distributive, we can assert that coz L is a base. This follows fromProposition 4.1.4, below, since when L is distributive, Spec Max L is a subspace ofSpec L . We close this section by making a connection to work of Martinez and Zenk. Acomplete lattice that is isomorphic to Id L for some distributive join-semilattice L iscalled an algebraic frame . (We discuss join-semilattices that satisfy the distributivitycondition in detail in Section 4, below.) In [MZ], Martinez and Zenk define a Yosidaframe to be an algebraic frame in which every compact element is the infimum of18he maximal elements above it. They define a frame F to be finitely subfit if: for allcompact elements b (cid:54)≤ a in F , there is some w ∈ F such that a ≤ w < w ∨ b = 1.One of the main results of [MZ] is their Proposition 4.2: Suppose A is an algebraicframe with F IP (i.e., the meet of any two compact elements is compact). Then A isYosida if and only if it is finitely subfit . To translate this into our terminology, A isa finitely subfit algebraic frame if and only if A is isomorphic to the lattice of idealsof some ideally conjunctive distributive join-semilattice. With this observation, wecan see that a much stronger statement than [MZ] Proposition 4.2 is an immediatecorollary of our Proposition 3.3.1. Corollary 3.4.1.
Let L be a join-semilattice. Every principal ideal of L (i.e., everycompact element of Id L ) is the meet of the maximal ideals that contain it if and onlyif L is ideally conjunctive. Observe that the ideals of a lattice are defined entirely in terms of the join oper-ation. In other words, if F is the forgetful functor from lattices to join-semilattices,and L is a lattice, then Id L = Id F L . Therefore, the corollary applies to all algebraicframes, and of course much more. The Martinez-Zenk result is an immediate conse-quence. The corollary shows that the FIP hypothesis is unnecessary. In fact, thereis no need to refer to meets, except as a way to restate the condition in Proposition3.3.1. Nor are any hypotheses about distributivity needed.
1. Let wL be the sublattice of W L generated by { coz a | a ∈ L } . As a sublatticeof the power set of max L , wL is distributive. Can we characterize the rela-tionship between L and wL algebraically? What is the relationship betweenSpec Max L and Spec wL ? (We answer these questions in subsection 5.2.)2. Since the criterion for { coz b | b ∈ B } to cover coz a is not finite, coz a in generalis not compact. For example, let L be the join semilattice of all open subsets[0 , ii ), Spec Max L is homeomorphicwith [0 ,
1] and the cozero sets are open subsets of [0 , a compact? Characterize those L for which coz a is compact for all a ∈ L .3. The conjunction condition for join-semilattices is first-order in the languageof join-semilattices, but though it is equivalent to the higher-order conditionthat every principal ideal is an intersection of maximal ideals. Is there a first-order way of stating the conjunction condition for morphisms (Definition 3.2.1)?(This must avoid direct reference to maximal ideals.)19 Distributive join semilattices
A join-semilattice L is said to be distributive if ∀ a, b , b ∈ L : a ≤ b ∨ b ⇒ ∃ a , a ∈ S s.t. a ≤ b & a ≤ b & a = a ∨ a .The distributive join semilattices are important for two reasons. First, these arethe most general semilattices for which there is a good topological representationtheory; see [G]. Second, the compact elements of an algebraic frame form a distribu-tive join semilattice with 0 and every distributive join semilattice arises this way.Distributivity proves to be a powerful but subtle property. In the present subsection, we review results from [G] concerning the relationship ofa distributive join-semilattice L to its frame F = Id L of join-semilattice ideals. Ina few cases, we supply details that are relevant to present work and are not fullyelaborated in [G]. We provide specific references to the relevant content of [G].Recall that a frame F is a complete lattice with 0 and 1 in which the binary meetoperation ∧ distributes over any join: a ∧ (cid:95) B = (cid:95) { a ∧ b | b ∈ B } . A frame morphism is a map between frames that preserves ∧ and (cid:87) . An element c ∈ F is compact if for all subsets B of F , c ≤ (cid:87) B implies c ≤ (cid:87) B (cid:48) for some finite B (cid:48) ⊆ B . The set of compact elements of F is a ∨ -semilattice and is denoted cpt F .A frame is said to be algebraic if every element is a join of compact elements. Lemma 4.1.1.
Let L be a join semilattice, and let Id L denote the complete latticeof ideals of L . The following are equivalent: ( i ) L is distributive. ( ii ) Id L is a distributive lattice. ( iii ) Id L is a frame, i.e., it satisfies the infinite distributive law:for all I ∈ Id L and J ⊆ Id L , I ∧ (cid:95) J = (cid:95) { I ∧ J | J ∈ J } . roof. ( i ⇔ ii ) is Lemma 184 of [G]. ( iii ⇒ ii ) is obvious. ( i ⇒ iii ). Let J be a setof ideals. The distributivity of L implies that if a ≤ b ∨ · · · ∨ b n , then a = a ∨ · · · ∨ a n for some a i ≤ b i . Accordingly, each element of (cid:87) J is of the form (cid:87) A , where A isa finite subset of (cid:83) J . Using this, we show the infinite distributive law. I ∩ (cid:87) J consists of the (cid:87) A that are in I , but if (cid:87) A ∈ I , then each a ∈ A belongs to I ∩ J for some J ∈ J . Thus, I ∩ (cid:87) J ⊆ (cid:87) { I ∧ J | J ∈ J } . The other containment isobvious.Suppose L is a distributive join semilattice and Id L is its frame of ideals. Clearly,every J ∈ Id L is a supremum of principal ideals. Moreover, J ∈ Id L is compact ifand only if it is principal. Thus, Id L is algebraic. Proposition 4.1.2.
If a frame F is algebraic, then cpt F is a distributive join semi-lattice, and F ∼ = Id cpt F .Remark. The first assertion is a point-free version of (part of) Theorem 191 of [G].For the convenience of the reader, we include the proof. Note that the ∨ operation of F always restricts to a ∨ operation on cpt F . The ∧ operation of F does not—cpt F is not in general closed under meets. Proof.
Given a, b , b ∈ cpt F with a ≤ b ∨ b , we must find a , a ∈ cpt F with a ≤ b and a ≤ b , and a = a ∨ a . Since F is join-generated by cpt F , wehave: a ∧ b i = (cid:87) A i for some A i ⊆ cpt F , i = 0 ,
1. Using the operations in F , a = ( a ∧ b ) ∨ ( a ∧ b ) = (cid:87) A ∨ (cid:87) A . Since a is compact, there are finite sets A (cid:48) ⊆ A and A (cid:48) ⊆ A such that a = (cid:87) A (cid:48) ∨ (cid:87) A (cid:48) . The elements a i := (cid:87) A (cid:48) i , i = 0 , F and satisfy the required conditions. The second assertion follows fromthe observation that the following maps are inverses of one another:Id cpt F (cid:51) I (cid:55)→ (cid:95) I ∈ F, and F (cid:51) f (cid:55)→ ( ↓ f ∩ cpt F ) ∈ Id cpt F. The following lemma and proposition give the main features of the topologicalrepresentation theorem for distributive join-semilattices.
Lemma 4.1.3.
Suppose L is a distributive join semilattice. Let F ⊆ L be a filter, andsuppose I is an ideal maximal disjoint from F . Then I is prime (i.e., its complementis a filter).Proof. Suppose a, b (cid:54)∈ I . We must show that a and b have a lower bound that is notin I . Since I is maximal disjoint from F , a ∨ p ∈ F and b ∨ q ∈ F for some p, q ∈ I .21ince F is a filter, there is c ∈ F such that c ≤ a ∨ p and c ≤ b ∨ q . By distributivity, c = a (cid:48) ∨ p (cid:48) , with a (cid:48) ≤ a and p (cid:48) ≤ p . But a (cid:48) ≤ c ≤ b ∨ q , so by distributivity again a (cid:48) = b (cid:48) ∨ q (cid:48) , with b (cid:48) ≤ b and q (cid:48) ≤ q . Now b (cid:48) is a lower bound for both a and b .Moreover, b (cid:48) / ∈ I , since b (cid:48) ∨ q (cid:48) ∨ p (cid:48) = c (cid:54)∈ I , while q (cid:48) ∨ p (cid:48) ∈ I .Note that in the proof above, we may take p = q , since p ∨ q ∈ I . Let us examinehow the proof runs in the special case when F = { } . Suppose I is maximal and a, b (cid:54)∈ I . Then a ∨ p = 1 = b ∨ p for some p ∈ I . We apply distributivity once: since a ≤ b ∨ p , we have a = b (cid:48) ∨ p (cid:48) with b (cid:48) ≤ b and p (cid:48) ≤ p . So, b (cid:48) ≤ a and b (cid:48) ≤ b . Moreover,1 = a ∨ p = b (cid:48) ∨ p (cid:48) ∨ p = b (cid:48) ∨ p (cid:54)∈ I . Since p ∈ I , b (cid:48) (cid:54)∈ I . Proposition 4.1.4.
Suppose L is a distributive join semilattice. Let Spec L denotethe set of prime ideals of L with the topology generated by sets spec a := { p ∈ Spec L | a (cid:54)∈ p } , where a ∈ L . Then the map J (cid:55)→ { p ∈ Spec L | J (cid:54)⊆ p } is a frame-isomorphism of Id L with the topology of Spec L .Proof. This is a restatement of [G], Lemma 186.
We have shown that in a distributive join semilattice, all maximal ideals are prime.Suppose L is a conjunctive join semilattice that is not distributive. Does L have amaximal ideal that is not prime? In this subsection, we answer in the affirmative forcomplete (and in particular, for finite) join semilattices.Let L be a finite conjunctive join semilattice. We can represent L as a T subbaseon a finite set X . Spec Max L is identified with X in the discrete topology. Each point x ∈ X is identified with the maximal ideal m x := { a ∈ L | x (cid:54)∈ a } .Suppose L fails to be distributive. Then there are a, b, c ∈ L such that: a ⊆ b ∪ c and ∀ b (cid:48) , c (cid:48) ∈ L, b (cid:48) ⊆ a ∩ b & c (cid:48) ⊆ a ∩ c ⇒ b (cid:48) ∪ c (cid:48) (cid:54) = a .(Note that a ∩ b and a ∩ c need not be in L .) Since X is finite, there is a largest b (cid:48) ∈ L such that b (cid:48) ⊆ a ∩ b and a largest c (cid:48) ∈ L such that c (cid:48) ⊆ a ∩ c .22ow, a (cid:54) = b (cid:48) ∪ c (cid:48) , so b (cid:48) ∪ c (cid:48) does not contain either :( i ) some x ∈ a \ c , or ( ii ) some y ∈ a ∩ b ∩ c , or ( iii ) some z ∈ a \ b . xyz......... bca In case ( i ), a and b contain x , i.e., are not in m x , but any lower bound in L for a and b does not contain x , and therefore is in m x . Case ( iii ) is similar: a and c are not in m z , but any lower bound in L for a and c is in m z . In case ( ii ), a , b and c all fail tobelong to m y , but the only lower bounds for a and b in L are in m y , and similarlyfor a and c . Thus, we have shown that any finite conjunctive join semilattice thatfails to be distributive has a maximal ideal that is not prime. The argument used above in the finite case depends only on the fact that everysubset of L has a least upper bound. Thus, we may replace “finite” with “complete”in the assertion proved above. Proposition 4.2.1.
Any complete conjunctive join semilattice that fails to be dis-tributive has a maximal ideal that is not prime.
1. Rhodes [R] mentions that an epimorph of a distributive join semilattice neednot be distributive. State useful conditions on L and an up-set U (possibly { } ) that are equivalent to L/R U being distributive.2. Is it true in general (i.e., without the completeness assumption) that a con-junctive join semilattice that fails to be distributive has a maximal ideal thatis not prime?3. Examine the relationship between the representation theory for distributivejoin-semilattices and the representation theory for conjunctive join-semilatticeswhen both hypotheses are satisfied. Let X be a topological space. We call a base B for the topology of X annular if ∅ and X are in B and B a sublattice of the frame of open sets of X . A Wallman base or X is an annular base B such that:If u ∈ U ∈ B , then there exists V ∈ B with u (cid:54)∈ V and U ∨ V = X . Lemma 5.1.1.
Let B be an annular base for X . Then X is a Wallman base if andonly if, for each u ∈ X , the ideal m u := { V ∈ B | u (cid:54)∈ V } is maximal. A T space has a Wallman base if and only if it satisfies the T separation axiom;see [GM]. We generally require T . In any case, we will always be explicit aboutseparation assumptions.If B is an annular base containing { x } c (the complement of the point x ) for each x ∈ X , then evidently X is T (points are closed) and B is Wallman. Lemma 5.1.2.
Let B be an annular base for a compact T space X . Then B isWallman.Proof. Suppose u ∈ U ∈ B\ m u . We can cover X with U together with neighborhoods V x of each x (cid:54)∈ U , with each V x not containing u . Finitely many V x will do, and theirunion, call it V , also misses u . Moreover, U ∪ V = X .Recall that B is conjunctive if for any W and U in B with W (cid:40) U , there is V ∈ B such that V ∪ W (cid:40) X = V ∪ U . Lemma 5.1.3.
Let X be a topological space. Any Wallman base for X is conjunctive.Proof. Given W (cid:40) U in B , select a point u ∈ U that does not belong to W . Usingthe Wallman assumption, select V ∈ B with U ∪ V = X and u (cid:54)∈ V . Then V ∪ W (cid:40) X = V ∪ U .The converse is not true. A conjunctive annular base for the topology of R neednot be a a Wallman base. This is shown by the second example below. The firstexample does not accomplish this goal, but we include it because it is useful inunderstanding the second Example 5.1.4.
In this example, let B be the sublattice of the topology of R con-taining R and ∅ and generated by the finite open intervals as well as the sets of theform E ∪ U , where E = ( a, ∞ ) or E = ( −∞ , b ) for some a, b ∈ R , and U is an openinterval containing 0. The interval U = ( − , ∞ ) is in B \ m , but there is no V ∈ m such that V ∪ U = R . So this is not a Wallman base. This example falls short ofour goal, however, because B is not conjunctive. Let W = ( − , ∪ (0 ,
1) and let U = ( − , V ∪ U = R , then 0 ∈ V , so V ∪ W = R .24 xample 5.1.5. We shall modify the previous example by demanding that thefinite open intervals in B must either contain 0 or not have 0 as an endpoint. Thus,if 0 (cid:54)∈ W ∈ B , then 0 has a neighborhood in B that misses W . Proposition 5.1.6. B of Example 5.1.5 is not Wallman, yet it is conjunctive.Proof. As in the first example, the interval U = ( − , ∞ ) is in B \ m , but there isno V ∈ m such that V ∪ U = R , so this is not a Wallman base. To show that B is conjunctive, suppose W, U ∈ B and W (cid:40) U ( W is a proper subset of U ).We shall show there is V ∈ B such that V ∪ W (cid:40) R and V ∪ U = R . Suppose U = ( a , b ) ∪ ( a , b ) ∪ · · · ∪ ( a n , b n ) and W = ( a (cid:48) , b (cid:48) ) ∪ ( a (cid:48) , b (cid:48) ) ∪ · · · ∪ ( a (cid:48) n (cid:48) , b (cid:48) n (cid:48) ). Herewe have( i ) −∞ ≤ a < b ≤ a < b ≤ · · · < b n ≤ ∞ ;( ii ) for all i , a i (cid:54) = 0 and b i (cid:54) = 0;( iii ) if a = −∞ or b n = ∞ , then for some i , a i < < b i .The same conditions apply to W . Moreover, each interval in the decomposition of W is contained in some interval in the decomposition of U . We might, for example,have ( a i , b i ) ⊇ ( a (cid:48) j , b (cid:48) j ) ∪ ( a (cid:48) j +1 , b (cid:48) j +1 ), with a i = a (cid:48) j , b (cid:48) j = a (cid:48) j +1 and b i = b (cid:48) j +1 . In otherwords ( a i , b i ) \ W is a single point, b (cid:48) j . Now for any (cid:15) >
0, let V := V ( (cid:15), U ), bedefined as follows:1. a = −∞ and b n = ∞ . Then V will be the union( b − (cid:15), a + (cid:15) ) ∪ ( b − (cid:15), a + (cid:15) ) ∪ · · · ∪ ( b m − − (cid:15), a n + (cid:15) ) . a > −∞ and b n = ∞ . Then V will be the union( −∞ , a + (cid:15) ) ∪ ( b − (cid:15), a + (cid:15) ) ∪ ( b − (cid:15), a + (cid:15) ) ∪ · · · ∪ ( b m − − (cid:15), a n + (cid:15) ) ∪ ( − (cid:15), (cid:15) ) . a = −∞ and b n < ∞ . Then V will be the union( b − (cid:15), a + (cid:15) ) ∪ ( b − (cid:15), a + (cid:15) ) ∪ · · · ∪ ( b m − − (cid:15), a n + (cid:15) ) , ( b n − (cid:15), ∞ ) ∪ ( − (cid:15), (cid:15) ) . a > −∞ and b n < ∞ . Then V will be the union( − (cid:15), (cid:15) ) ∪ ( −∞ , a + (cid:15) ) ∪ ( b − (cid:15), a + (cid:15) ) ∪ · · · ∪ ( b m − − (cid:15), a n + (cid:15) ) ∪ ( b n − (cid:15), ∞ ) . W is a proper subset of U , some of the endpoints { a (cid:48) j , b (cid:48) j } defining the intervalsof W are in the interior of the intervals of U , or W is the union of a subset of theintervals defining U . Moreover, if W does not contain 0, then it misses an intervalabout 0. Thus, in any case, we may choose (cid:15) > W -intervals that are not endpoints of U -intervals belong to V ( (cid:15), U ). This assuresthat V ( (cid:15), U ) ∪ W (cid:54) = R . By construction V ( (cid:15), U ) ∪ U = R .Observe that { U ⊆ R | U open and 0 (cid:54)∈ U } is a maximal ideal in the full topologyof R . In contrast, in the example above, m := { U ∈ B | (cid:54)∈ U } fails to be a maximalideal of B . It is contained in I fin , the ideal of all finite unions of finite open intervalswith non-zero endpoints. Moreover, I fin itself is not maximal. It is contained inthe ideal m + ⊆ B whose elements are all finite unions of intervals of the form ( a, b )where 0 (cid:54) = a ∈ {−∞} ∪ R and 0 (cid:54) = b ∈ R , with the proviso that any element of m + containing an interval of the form ( −∞ , b ) must also contain an interval ofthe form ( − (cid:15), (cid:15) ), (cid:15) >
0. We also have the ideal m − , which is defined analogously.Both these ideals are maximal. What we have here, then, can be viewed as the realline with two additional points added, call them 0 + and 0 − . Since m ⊆ m + and m ⊆ m − , each of the new points belongs to the closure of 0. A neighborhood basefor 0 + consists of the elements in the complement of m + , namely all finite unions ofintervals that contain a set of the form ( − (cid:15), (cid:15) ) ∪ ( x, ∞ ), and analogously for 0 − . Eachhas a neighborhood that does not contain the other, but they do not have disjointneighborhoods. Proposition 5.1.7 ([J82] IV.2.4) . If B is a Wallman base for a T -space X , then η B : X → Spec
Max B is an embedding with dense image. In the proposition as presented in [J82], Spec
Max B refers to the subspace theprime spectrum Spec B consisting of the maximal ideals. If we treat B as a joinsemilattice and construct Spec Max B as above, then we get the same space. Thereason is that the definition of ideal we used in the context of join semilattices is thesame as the definition used for distributive lattices. The same is true of maximalideals, and in both settings the definition of the topology is the same. For any join-semilattice L with 1, there is a 1- ∨ -preserving morphism d L : L → dL that is universal to distributive lattices with 1. In this subsection, we consider thisconstruction when L is conjunctive. By the universal mapping property, there is a1- ∨ - ∧ -preserving surjection w L : dL → wL , where w L is the distributive sub-lattice26f the topology of Spec Max L that is generated by { coz a | a ∈ L } . We providean example showing that w L may fail to be injective, and we prove that in general wL ∼ = dL/R . Throughout the remainder of this section, all join-semilattices and all distributivelattices have , and all morphisms preserve . Let L be a join semilattice. Wesay that dL is a free distributive lattice over L if dL is a distributive lattice, andthere is a 1- ∨ -morphism d L : L → dL such that for any 1- ∨ -morphism f : L → B ,with B a distributive lattice, there is a unique 1- ∨ - ∧ -morphism f : dL → B suchthat f = f d L . The universal mapping property guarantees uniqueness up to uniqueisomorphism, so we speak of the free distributive lattice with 1 over L . The existenceof dL follows from the fact that distributive lattices form a varietal category. In orderto examine examples, we provide a concrete description.We may construct dL by imitating the construction in [J82]. Let P L denote thepower set of L . Define δ L : L → P L by δ L ( a ) := ↑ a . Let dL to be the sublattice of P L genertated by the image of L . We endow dL with the opposite of the natural order,so for subsets U, V ⊆ L , U ≤ V means V ⊆ U , U ∨ V := U ∩ V and U ∧ V := U ∪ V .(In other words, we are viewing dL as a sub-poset of ( P L ) op .) Then d L is a 1- ∨ morphism. A slight modification of the proof of [J82], II.1.2 shows that δ L has theuniversal mapping property required in the definition of the free distributive latticeover L .Below, we shall use an equivalent representation dL as a sublattice of P L withthe natural order, U ≤ V ⇔ U ⊆ V . We simply define d L ( b ), b ∈ L , to be thecomplement of ↑ b , i.e., d L ( b ) := { y ∈ L | b (cid:54)≤ y } ∈ P L. (1)If L is conjunctive, b (cid:54)≤ y if and only if y is contained in a maximal ideal missing b ;therefore, d L ( b ) is the set of all y ∈ L such that (cid:98) y vanishes at some element of coz b .The join in dL is set-theoretic union in P L . As required, d L is join-preserving: d L ( a ) ∨ d L ( b ) = { y ∈ L | a (cid:54)≤ y or b (cid:54)≤ y, } = { y ∈ L | a ∨ b (cid:54)≤ y } . (2)Meets in dL are described as follows: d L ( b ) ∧ · · · ∧ d L ( b n ) = { y ∈ L | b (cid:54)≤ y } ∩ · · · ∩ { y ∈ L | b n (cid:54)≤ y } (3)= { y ∈ L | b (cid:54)≤ y & · · · & b n (cid:54)≤ y } (4)If we assume L is conjunctive, d L ( b ) ∧ · · · ∧ d L ( b n ) is the set of y ∈ L such that (cid:98) y vanishes at some point of coz b i for each i = 1 , . . . , n . Note that it is not requiredthat (cid:98) y vanish at some point of the intersection of all these cozero sets.27ow, let L be a conjunctive join semilattice and let wL denote the sublattice ofthe topology of Spec Max L that is generated by { coz a | a ∈ L } . For compatibilitywith other notation, we used the notation w L : L → wL , with w L ( a ) := coz a . Thus, w L ( b ) ∧ · · · ∧ w L ( b n ) = coz b ∩ · · · ∩ coz b n (5)= { m ∈ Spec
Max | b (cid:54)∈ m & · · · & b n (cid:54)∈ m } . (6)By the universal mapping property of d L , there is a unique 1- ∨ - ∧ -morphism w L : dL → wL such that w L d L = w L . Evidently, we must have w L (cid:0) { y ∈ L | b (cid:54)≤ y & · · · & b n (cid:54)≤ y } (cid:1) = { m | b (cid:54)∈ m & · · · & b n (cid:54)∈ m } . This need not be an injective map, as we show below by example.
Example 5.2.1.
Let L be any set containing 1, with join operation defined by a ∨ b = 1 for any distinct elements a and b of L . This is clearly conjunctive. We have d L ( b ) = L \ { , b } , and d L ( b ) ∧ · · · ∧ d L ( b n ) = L \ { , b , . . . b n } . The maximal idealsof L are the singletons { b } , with b (cid:54) = 1. We have w L ( b ) = coz b = {{ x } ∈ Spec
Max L | x (cid:54) = b } . The map w L is given by w L (cid:0) L \ { , b , . . . b n } (cid:1) = (cid:0) Spec
Max L (cid:1) \ { { b } , . . . , { b n } } . In this example, w L is injective.Let L be a finite conjunctive join semilattice, and let X = Spec Max L . We view L as a sub-join-semilattice of P X . Since X is finite and it’s topology is T , for each x ∈ X , { x } c = X \ { x } ∈ L . The maximal ideal x is the downset of { x } c . We havethat W L = wL = P X . Example 5.2.2.
Suppose X = { a, b, c } and L is the join semilattice that consistsof the non-empty elements of P X . We write the subsets of X as strings: a := { a } , ab := { a, b } , etc. Thus, L = { a, b, c, ab, ac, bc, abc } , and m a := ↓ bc = { b, c, bc } . Thefollowing diagrams show the structure of dL . To avoid clutter, we use i , j , and k to represent a , b and c taken in any order. The diagram on the left represents theelements as meets of the generators, while the one on the right shows the correspond-ing elements as downsets of L . Each node on the right labeled with i , j , and/or k represents three distinct elements of dL . Accordingly, the cardinality of d L is 18.Since the cardinality of wL is 8, evidently, w L is not injective.28 L ( ijk ) d L ( ij ) d L ( ij ) ∧ d L ( ik ) d L ( i ) d L ( ij ) ∧ d L ( ik ) ∧ d L ( jk ) d L ( i ) ∧ d L ( jk ) d L ( i ) ∧ d L ( j ) ∅ { ab, ac, bc, a, b, c }{ ik, jk, i, j, k }{ jk, i, j, k }{ jk, j, k, } { a, b, c }{ j, k }{ k }∅ Proposition 5.2.3.
Let L be a conjunctive join-semilattice. The congruence on dL induced by w L : dL → wL is R ( dL ) , and hence wL ∼ = dL/R .Proof. First, by the same arguments used in the proof of Lemma 3.1.9, we have thatSpec
Max wL = Spec Max L . By Lemma 3.1.1, wL is conjunctive. We now show that w L − (1 W L ) = { dL } . Suppose q ∈ dL \ { } . We show that w L ( q ) (cid:54) = 1. Since d L : L → dL is a 1- ∨ -morphism and dL is distributive, q = (cid:87) ni =1 d L ( b i ), with b i ∈ L \ { } .Thus, q ≤ d L ( b i ), so w L ( q ) ≤ w L d L ( b i ) = coz b i <
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