A class of infinite convex geometries

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.


Introduction
There are various ways to characterize finite convex geometries; see Chapter 3 of our [4], which combines results from Dilworth [9], Edelman and Jamison [11], Duquenne [10], and Monjardet [15]. These characterizations can be either combinatorial or lattice theoretical in nature.
Infinite versions of convex geometries occur in several sources, each of which is equivalent to a closure operator with the anti-exchange property, plus some finiteness conditions to provide structure. Crawley and Dilworth [7] consider dually algebraic, dually coatomic, locally distributive lattices. Adaricheva, Gorbunov and Tumanov [3] discuss closure operators with the anti-exchange property whose closure lattices are weakly atomic, dually spatial and atomistic. Adaricheva and Nation [4] are concerned with algebraic closure operators with the antiexchange property. See also Nakamura and Sakaki [16], Adaricheva and Pouzet [5], and Adaricheva [2].
Here we consider a fourth type, inspired by Gorbunov's classic result [12]: Every element of a complete lattice L has a canonical join decomposition if and only if L is strongly coatomic, lower continuous, and join semidistributive. Adding lower semidmodularity to the list will give us an infinite class of convex geometries; see Theorem 7.
2. The anti-exchange property Definition 1. A closure system (X, γ) satisfies the anti-exchange property if for all x = y and all closed sets A ⊆ X, x ∈ γ(A ∪ {y}) and x / ∈ A imply that y / ∈ γ(A ∪ {x}).
Date: January 20, 2015. 1 Equivalently, a closure operator satisfies the anti-exchange property if for all closed sets A ⊆ X and elements x, Examples of closure operators with the anti-exchange property include • the convex hull operator on Euclidean space E n , • the convex hull operator on an ordered set, • the subalgebra-generated-by operator on a semilattice, • the algebraic-subset-generated-by operator on a complete lattice. For a closure system (X, γ), we will let Cld(X, γ) denote the lattice of γ-closed subsets of X. A closure system is zero-closure if γ(∅) = ∅.

Definition 2.
A zero-closure system that satisfies the anti-exchange property is called a convex geometry.
(This common convention is a bit awkward, as some useful closure operators with the anti-exchange property have a non-empty closure of the empty set. Nonetheless, we shall retain it.) A lattice is strongly coatomic if a < c in L implies that there exists b such that a ≤ b ≺ c. A closure system is strongly coatomic if its lattice of closed sets is so.
Theorem 3. For a strongly coatomic closure system (X, γ), the following are equivalent.
The equivalence of the preceding theorem is also valid for algebraic closure systems [4] and [2].

SCLCC lattices
A complete lattice is lower continuous if for every down-directed set D ⊆ L, a ∨ D = d∈D (a ∨ d). We denote a strongly coatomic, lower continuous, complete lattice as SCLCC, and a closure system is SCLCC if its lattice of closed sets is so. Some basic properties of SCLCC lattices follow.
Lemma 4. Let L be an SCLCC lattice.
(1) Every nonzero join irreducible element is completely join irreducible.
(2) If w ≻ c in L, then there exists an element k that is minimal w.r.t. k ≤ w, k c, and each such element is join irreducible. (3) Every element of L is a (perhaps infinite) join of join irreducible elements.
A complete lattice in which every element is a join of completely join irreducible elements is said to be spatial. Clearly, this is a desirable property for any sort of "geometry." In [5], it was shown that every weakly atomic convex geometries is spatial. The preceding lemma says that SCLCC lattices have an even better property: a complete lattice is strongly spatial if it is spatial and every nonzero join irreducible element is completely join irreducible. The set of nonzero join irreducible elements of a lattice L will be denoted by Ji(L).
The next lemma is useful in connection with local distributivity. A complete lattice is coatomistic if every element is a meet of coatoms.
Lemma 5. A distributive, coatomistic SCLCC lattice is isomorphic to the boolean algebra of all subsets of its coatoms.

Join semidistributive SCLCC lattices
Next we generalize some equivalences of join semidistributivity which are well-known for finite lattices.
The implication is known as the join semidistributive law. In view of Lemma 4 and the results in Jónsson and Kiefer [13], we consider two potentially infinite versions of the law: It is an elementary exercise that join semidistributive, lower continuous, complete lattices satisfy SD ′ ∨ , and we shall use that at will.
We say that w = A is a canonical join decomposition if the join is irredundant, and w = B implies A ≪ B. Theorem 6. The following are equivalent for an SCLLC lattice L.
(2) L satisfies SD * ∨ . (3) Every element of L has a canonical join decomposition. (4) If w ≻ c in L, then there exists a join irreducible k which is the unique minimal element such that k ≤ w but k c.
. Assume that property (4) holds, and fix an element w ∈ L. Let C = {c ∈ L : w ≻ c}. For each c ∈ C, we can find an element k c such that k c ≤ w but k c c. We claim that w = c∈C k c canonically. Clearly c∈C k c = w, since each k c is below w, while the join is below no lower cover of w. If It follows that the join is irredundant.
Suppose w = A and consider c ≺ w. There exists some a 0 ∈ A such that a 0 c, though a 0 ≤ w, whereupon k c ≤ a 0 . Since this holds for all c ∈ C, we have {k c : c ∈ C} ≪ A, as desired.
(3) ⇒ (2). Suppose that w = S = T . If there is a canonical join decomposition w = U in L, then U refines both S and T , so that for each u ∈ U there exist s ∈ S with u ≤ s, and t ∈ T with u ≤ t. Hence each u ≤ s ∧ t for some pair, and it follows that w = (s ∧ t).

Characterizations of LCSCC convex geometries
First, we introduce a few more terms. A strongly coatomic complete lattice is said to be locally distributive (or lower locally distributive or meet distributive) if for any x ∈ L the interval [µ(x), x] where µ(x) = {y : y ≺ x} is a distributive lattice (and hence for SCLCC lattices a boolean algebra). For The set of extreme points of A is denoted Ex(A). In lattice terms, for a strongly spatial lattice L, we identify the element a with the set Ji(a) = {p ∈ Ji(L) : p ≤ a}. Then x ∈ Ji(a) is an extreme point of a if a > (Ji(a) \ {x}). This means that (i) x is join prime in the ideal id(a), and (ii) there is no other join irreducible y with x < y ≤ a.
Recall that a lattice L is called lower semimodular We now extend some characterizations of finite convex geometries to SCLCC geometries. In this setting we want to think of a lattice in terms of its standard representation as a closure system on its set of join irreducibles. Note that properties (1)-(2) of the next theorem are about closure systems, while (3)-(6) are lattice properties. For the finite case, various parts of the theorem are due to R.P. Dilworth, P. Edelman and R. Jamison, and V. Duquenne.
Theorem 7. Let L be an SCLCC lattice. Then the following are equivalent.
(1) L is the closure lattice Cld(X, γ) of a closure space (X, γ) with the AEP. Proof. The equivalence of (1) and (2) is Theorem 3. To see that (2) (3), that L is locally distributive and lower semimodular, and we want to show that L is join semidistributive by means of property (4) of Theorem 6, which yields (4) of this theorem.
So let w ≻ c, and let a be minimal w.r.t. a ≤ w, a c. Suppose by way of contradiction that there exists another such element b. Choose an element b ′ with b ≤ b ′ ≺ a ∨ b. Using strong coatomicity, form a (possibly transfinite) sequence a α for α ≤ κ such that (i) a 0 = a ∨ b and a κ = a, (ii) a α+1 ≺ a α , (iii) a λ = α<λ a α for limit ordinals λ. We show by induction that (a α ∧ b ′ ) ∨ c = w for all α ≤ κ, so that a α ∧ b ′ c.
If the statement holds for a α , then by lower semimodularity, a α+1 , a α ∧ b ′ and a α ∧ c are three distinct lower covers of a α . Hence by local distributivity a α+1 ∧ (a α ∧ b ′ ) a α ∧ c, whereupon a α+1 ∧ b ′ c and (a α+1 ∧ b ′ ) ∨ c = w. At limit ordinals, the claim follows by lower continuity.
For α = κ, this contradicts the minimality of a. Thus L is join semidistributive.

Sublemma 8.
In an SCLCC lattice, w = Ex(w) iff for each element c with c ≺ w, there exists a unique j c in Ji L such that j c ≤ w and j c c.
Proof. Assume w = Ex(w), and let c ≺ w. There exists j ∈ Ex(w) such that j c, and j (Ji(w) \ {j}) because j is extreme. In that case, the latter join must in fact be c, since c is a join of join irreducibles below w. Hence j is unique.
Conversely, suppose that for each c ≺ w such a j c exists. Then {j c : c ≺ w} = w, since the join is below w but not below any of its lower covers. On the other hand, each such j c is extreme, as c = (Ji(w) \ {j c }), since j c is the only join irreducible below w that not below c. Now assume (4), that L is join semidistributive and lower semimodular. Let w ≻ c in L. Then SD ∨ and lower continuity give us the existence of a join irreducible element j that is the unique minimal element w.r.t. j ≤ w, j c. If j < y ≤ w, then by lower semimodularity y ≻ y ∧ c, whence y = j ∨ (y ∧ c) is join reducible. Thus j is an extreme element for w.
The equivalence of (5) and (6), and that (6) implies (2) for the standard representation of a spatial lattice L as a closure operator on its join irreducibles, are both easy.
A lattice L is called atomistic if every nonzero a ∈ L is a join of atoms. Atomistic convex geometries were characterized in Proposition 3.1 of Adaricheva, Gorbunov and Tumanov [3]. For the SCLCC case, the proof is particularly easy.
Corollary 9. Any SCLCC atomistic join semidistributive lattice is the closure lattice of some convex geometry.
Proof. It is enough to show that a SCLCC atomistic join semidistributive lattice is lower semimodular. Indeed, it follows from join semidistributivity that if a ≺ b, then there exists a unique atom t such that t ≤ b, t a. If c ≤ b, then either t ≤ c and thus c and c ∧ a differ by the single atom t, so that c ∧ a ≺ c, or else t c in which case every atom below c is below a, whence c ≤ a.

Examples of SCLCC convex geometries
There are natural examples of the kind of geometries described in Corollary 9, obtained by taking standard convex geometries and adding finiteness conditions to ensure strong coatomicity and lower continuity.
• If P is an ordered set such that both the ideal id(x) and the filter fil(x) are finite for every x ∈ P , then the lattice of convex subsets Co(P ) is an atomistic SCLCC convex geometry. • If S is a meet semilatiice such that the filter fil(x) is finite for every x ∈ S, then the lattice of subsemilattices Sub ∧ (S) is an atomistic SCLCC convex geometry. (We should note that in each example, the closure operator is algebraic, and an algebraic lattice is dually algebraic iff it is lower continuous.) For the first type, we know that the convex hull operator on an ordered set satisfies the AEP; it remains to show that if P has the property that every principal ideal and filter is finite, then Co(P ) is strongly coatomic and lower continuous.
Suppose that P has that property, and that A < B in Co(P ).
For lower continuity, it is convenient to use the equivalent formulation in terms of chains; see e.g. Theorem 3.8 of [17]. Let X be convex, and let C = {C i : i ∈ I} be a chain of convex subsets of P . We want to show that i (X ∨ C i ) ⊆ X ∨ i C i . Let w be in the left-hand side, and w.l.o.g. w / ∈ X. Either there exists a co-initial subset D = {C j : j ∈ J} for some J ⊆ I, and elements x j ∈ X and y j ∈ C j , such that x j ≤ w ≤ y j for all j ∈ J, or dually; assume the first. Because fil(w) is finite, there exists a co-initial subset E = {C k : k ∈ K} with K ⊆ J such that y k = y k ′ for all k, k ′ ∈ K. Then y k ∈ k∈K C k = i∈I C i , and fixing any k 0 ∈ K we have x k 0 ≤ w ≤ y k 0 . Thus w is in the right-hand side, and Co(P ) is lower continuous. Now consider Sub ∧ (S) for a meet semilattice S. Again, we know that the subsemilattice operator satisfies the AEP, and it remains to show that if S has the property that every principal filter is finite, then Sub ∧ (S) is strongly coatomic and lower continuous.
To show lower continuity, let X be a subsemilattice and let C = {C i : i ∈ I} be a chain of subsemilattices of S. Again, we want to show that i (X ∨ C i ) ⊆ X ∨ i C i , so let w be in the left-hand side. That means that for each i ∈ I there exist x i ∈ X and y i ∈ C i such that w = x i ∧ y i . Let x = i x i , which really is a finite meet. If x = w, we are done. Otherwise, we continue: because fil(w) is finite, there exists a co-initial subset E = {C k : k ∈ K} for some K ⊆ I such that y k = y k ′ for all k, k ′ ∈ K. Then w = x ∧ y k , and y k ∈ k∈K C k = i∈I C i , so w ∈ X ∨ i C i , as desired.
We can even combine these examples: if S is a meet semilattice in which every principal filter fil(x) is a finite tree, then the lattice of convex subsemilattices of S is an SCLCC convex geometry. See Adaricheva [1] and Cheong and Jones [6].
In a similar vein, if (P, ≤) is an ordered set in which every chain is finite and every interval is finite, then the lattice of suborders of ≤ on P is an SCLCC convex geometry. See Semenova [18].
Another construction yields SCLCC convex geometries that need not be atomistic. Our inspriration is the fact that a geometric lattice is isomorphic to the ideal lattice of its finite dimensional elements. (There is no chance for a similar characterization here, since for any non-limit ordinal α, the dual α d is an SCLCC convex geometry.) Our construction uses Jónsson and Rival's characterization of join semidistributive varieties [14].
Define certain lattice terms recursively: for k ≥ 0, Then consider the lattice inclusions These are equivalent to the corresponding identities x∨y k ≈ x∨(y ∧z). For example, SD ∨ (1) is equivalent to the distributive law.
Lemma 10. Let V be a lattice variety. Then every lattice in V is join semidistributive if and only if V satisfies SD ∨ (n) for some n < ω.
Let SD ∨ (n) be the variety of all lattices satisfying SD ∨ (n).
Theorem 11. Let L 0 be a lattice with the following properties.
• fil(x) is finite for each x ∈ L 0 .
• L 0 is lower semimodular. Then the filter lattice L = Fil(L 0 ) is an SCLCC convex geometry.
Proof. As usual, we order the filter lattice by reverse set inclusion: F ≤ G iff F ⊇ G. The filter lattice of any lattice is lower continuous and satisfies the equations of the original, in particular SD ∨ (n) in this case. It remains to show that L is strongly coatomic and lower semimodular.
Suppose F < G in L, i.e., F ⊃ G. Let k be an element maximal in F \ G, and note that k is meet irreducible. We claim that the filter generated by G ∪ {k}, say H = fil(G, k), satisfies F ≤ H ≺ G. Let ℓ be any element of H \ G. Then ℓ ≥ g ∧ k for some g ∈ G, and we may take g ≤ k * , where k * denotes the unique upper cover of k in L 0 . In that case, by lower semimodularity, g ≻ g ∧ k, whence also g ∧ k = g ∧ ℓ. It follows that H = fil(G, ℓ), and since ℓ is arbitrary, H ≺ G. Thus L is strongly coatomic.
The proof that L is lower semimodular is an adaptation of that for the corresponding dual claim in Theorem 11.1 of [17]. Assume that L is lower semimodular, and suppose that F ≺ F ∨ G = F ∩ G in Fil(L 0 ). Choose an element a maximal in F \G, and note that a is meet irreducible, thus by the finiteness of fil(a) completely meet irreducible. Then F = (F ∨ G) ∧ fil(a), and hence F ∧ G = fil(a) ∧ G. Let x be any element in (F ∧ G) \ G. Since x ∈ F ∧ G, there exists g ∈ G such that x ≥ a ∧ g. Because L is lower semimodular, a ∧ g ≺ a * ∧ g. On the other hand, every element of L is a meet of finitely many meet irreducibles, so x / ∈ G implies there exists a meet irreducible element b ≥ x with b / ∈ G. Now b ≥ a ∧ g and b g, so b ∧ g = a ∧ g, whence a ≥ b ∧ g. Thus fil(b) ∧ G = fil(a) ∧ G = F ∧ G; if follows a fortiori that fil(x) ∧ G = F ∧ G. As this holds for every x ∈ (F ∧ G) \ G, we have F ∧ G ≺ G, as desired.
So in particular, we could take L 0 to be the elements of finite depth in a direct product of finite convex geometries that satisfy SD ∨ (n) for some fixed n.

Discussion
In some sense, algebraic closure operators are the natural settings for any type of geometry. On the other hand, Crawley and Dilworth's setting of dually algebraic and strongly coatomic gives the nice equivalence of local distributivity and unique representations. Our hypothesis of SCLCC is slightly weaker, but means that we must assume lower semimodulariy along with local distributivity. The lattice (ω + 1) d × 2, with its atom doubled, shows that local distributivity by itself is not sufficient.