A Classification of Hull Operators in Archimedean Lattice-Ordered Groups With Unit

The category, or class of algebras, in the title is denoted by W. A hull operator (ho) in W is a reflection in the category consisting of W objects with only essential embeddings as morphisms. The proper class of all of these is hoW. The bounded monocoreflection in W is denoted B. We classify the ho’s by their interaction with B as follows. A “word” is a function w : hoW −→ W obtained as a finite composition of B and x a variable ranging in hoW. The set of these,“Word”, is in a natural way a partially ordered semigroup of size 6, order isomorphic to F(2), the free 0− 1 distributive lattice on 2 generators. Then, hoW is partitioned into 6 disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word (≈ F(2)). Of the 6: 1 is of size ≥ 2, 1 is at least infinite, 2 are each proper classes, and of these 4, all quotients are chains; another 1 is a proper class with unknown quotients; the remaining 1 is not known to be nonempty and its quotients would not be chains. * Corresponding author


Introduction
The present work seems to represent new theory imposed on the lattice F (2). This requires some background material from lattice-ordered groups ( -groups) and topology that we review here. For additional information, we refer the reader to [6], [1] and [14] for -groups, and [15], [18] for topology and C(X).
The following alternative description of a hull operator (in various classes of algebras C including W) is provided in [11]. A hull operator is an "essential closure operator" (with isomorphic objects identified), that is, where ≤ (respectively, e ≤) signifies embeds as a subobject (respectively, an essential subobject) and for which (i) hC = h(hC); (ii) if C e ≤ D ≤ hC, then hD = hC; (iii) hC is unique up to a unique isomorphism over C.
As demonstrated in [9], the collection of hull operators on W, hoW, is a complete lattice with a partial ordering defined "pointwise" as follows. For h 1 , h 2 ∈ hoW, h 1 ≤ h 2 means for every G ∈ W, G e ≤ h 1 G e ≤ h 2 G (up to unique isomorphism over G). The bottom of the lattice hoW is the identity operator Id, and the top is the essential completion hull operator e.
The category W is much informed by the Yosida Representation Theorem, which we now review.
For a Tychonoff space X (usually compact) define Note, D(X) with the pointwise order and addition is a lattice but is usually not a group (as addition is not always fully defined). We now recall the Yosida Representation Theorem from [23]. requisite to making η G (G) an -group; G and η G (G) are isomorphic Wobjects. Moreover, if φ : G −→ H is a W-homomorphism, that is, a lattice homomorphism for which φ(u G ) = u H , then there exists a unique continuous map τ : Y H −→ Y G, such that for every g ∈ G, η H (φ(g)) = η G (g) · τ . We suppress notation and write G ≤ D(Y G) and φ(g) = g · τ .
A partial view of hoW appears as a Hasse diagram in [11] pg. 167, in which are depicted several distinct chains, which are faithfully indexed by the regular cardinals. While hoW is huge (a proper class or larger, depending on one's definition of "proper class") for the purpose of providing distinction of present interests, we require, perhaps, only four examples, which we now describe, in terms of the Yosida Representation G ≤ D(Y G).
which are locally in G. The hull operator loc is a reflection in W, and Y locG = Y G. For additional, information see [24].
(2) The maximum essential reflection c 3 : is a filter base of dense open subsets of Y G and G −1 (R) δ , is the filter base of all countable intersections of elements of G −1 (R) (which are dense by the Baire Category Theorem). For any filter base of dense sets F on a space X, define and only if f 1 and f 2 agree on the intersection of their domains. Then, In [3]) the authors demonstrate that c 3 G is the maximum essential reflection in W. There is more information in many other places such as [5] and its references. For future reference, we note the following example. If G 0 is the eventually polynomial functions on N, then Y G 0 = αN, the one point compactification of N, c 3 G 0 = C(N) and Y c 3 G 0 = βN, theČech-Stone compactification of N.
(3) The essential completion e: Any compact Hausdorff space X has its "absolute" (Gleason cover, projective cover) which is an irreducible surjection π : aX −→ X, where aX is compact extremally disconnected. For G ∈ W one has Y G π ←− aY G and if one defines eG ≡ D(aY G) and φ : G −→ eG by φ(g) = π · g, then G e ≤ eG. There are details to this, of course, some of which is discussed in [9]. This is all a version of Conrad's description of the essential completeness and completion in [12]. The following observation is useful: D(aX) ≈ C[G (X) δ ], where G (X) δ is the filter base of dense G δ 's in X [16].
(4) The Dedekind completion of the divisible hull c: This is which is the -group ideal in eG that is generated by G (in [14], 54.23 and 57. 16).
Here is a preliminary result about the hull operators in Example 1.2. Proof. Observe that ≤ holds in (a) and (b). [ [24]] says that any W-object G, which is the "W-part" of an f -ring with identity has locG = G. As any c 3 -object is such a G, locG ≤ c 3 G. As to the loc ≤ c case, G = cG implies that G is projectable ( [14]) and projectable implies local ( [22]). Recall, if h 1 ≤ h 2 to prove h 1 < h 2 , it suffices to show that there exists an G ∈ W such that h 1 G < h 2 G. (a) Since any W-object G which is the "W-part" of an f -ring with identity has G = locG, G 0 from (2) in Example 1.2 satisfies G 0 = locG 0 < c 3 G 0 .
As to the other inequality in (a), if X is compact, then Y C(X) = X and C(X) = c 3 C(X). If X is also infinite, then D(aX) contains unbounded functions (as a compact infinite extremally disconnected space is not a Pspace [18]). Consequently, C(X) = c 3 C(X) < D(aX) = eC(X).
(b) loc < c. Since for every G, Y locG = Y G, whereas, Y cG = aY G, one may proceed as in (a) but with X compact, infinite and not extremally disconnected. On the other hand, since for every compact X, cC(X) = BeC(X) = C(aX) and, usually, C(aX) < D(aX), it follows that c < e.
(c) We demonstrate that there exists G 1 and G 2 such that c 3 G 1 < cG 1 and cG 2 < c 3 G 2 . If G 1 = C(X), for X compact, not extremally disconnected, then G 1 = c 3 G 1 < C(aY G 1 ) = cG 1 . Now take Y to be an infinite, extremally disconnected and pick g ∈ D(Y ) unbounded and g(y) > 0 for every y ∈ Y . Let G 2 be the sub--group of D(Y ) generated by C(Y ) and g. Then, cG 2 = {f ∈ D(Y ) | |f | ≤ ng for some n ∈ N and g ∈ G 2 } and We now review the bounded monocoreflection in W. For G ∈ W, with weak unit u G , define Then BW is a monocoreflective subcategory of W, where the functor B : W −→ BW is the monocoreflector. [11] investigates the interactions of B with hull operators with a quite different thrust than the present paper. Section 3 there lists various properties of B from which one, easily, infers the following.
For any G ∈ W, one has the following commutative diagram in which each arrow is an essential embedding: Here Bh (respectively, hB) is the composition of the functions h and then B (respectively, B and then h).
Throughout we make use of Proposition 1.4 without mention. The subject of hoC for various categories C, has a large literature. See the bibliographies of our previous papers [9], [10], and [11]. We won't totally replicate those bibliographies here but do mention Conrad's seminal papers [12], [13] and Martinez's survey [27]. There are various "nearly" dual situations, including coComp, covering operators for compact spaces (the largest being the "a" mentioned in (3) of Example 1.2). The connection between hoW and coComp are studied in [9] and [28]. By itself coComp is examined in [21] and [30], [29]. We apologize to authors not mentioned.

Words
Definition 2.1. (a) "x" is a variable ranging in hoW, and is a function from hoW −→ W W , whose action at x = h is the function h ∈ W W (whose action at G ∈ W is G → hG).
(b) The bounded monocoreflection B ∈ W W , and may be construed to be the constant function hoW −→ W W , whose action is h → B for every h. Then, the composition of functions w(x) = xB, x after B, is a function from hoW −→ W W , whose action at x = h is G → hBG. A (general) word w(x) or w is a function, which is a finite string of successive compositions, w(x) = y n · y n−1 · · · y 2 · y 1 , where n ∈ N, for 1 ≤ i ≤ n, y i = h or B and y i+1 is after y i for 1 ≤ i < n. Throughout, "Word" is the set of words, that is, such functions.
(c) Let w 1 , w 2 ∈ Word. The ordering w 1 ≤ w 2 is "pointwise" : for every where the last ≤ signifies "is embedded as a W-subobject of". The multiplication w 2 w 1 is composition of functions, w 2 after w 1 , which is expressible as concatenation of the strings. Note, the example w(x) = xB is the product Theorem 2.2. There are exactly 6 words, listed and ordered (per 2.1) as in the following picture in which the order is ↑ Consequently, Word is order isomorphic to F(2).
Before proving Theorem 2.2, we note that Word has more algebraic structure as described below. This is intriguing and could well inform some of the issues articulated in section 4. But, we don't know about that, so we shall omit the proof of Corollary 2.3 below (which we have only achieved through tedious and lengthy, case-by-case verifications).
A partially ordered semigroup is an algebraic system (S, ·, ≤) for which (S, ·) is a semigroup and (S, ≤) is a partially ordered set, and a ≤ b implies that ca ≤ cb ac ≤ bc for all c ∈ S, and is an -semigroup if (S, ≤) is a lattice and and is an -semigroup if also See [17], where it is shown that (∨) does not imply (∧). We have coined the term " ". We now prove Theorem 2.2.
Proof. The partial order and multiplication throughout this proof are as in 2.1. We shall use the following features of B and any h ∈ hoW.
(i) B is decreasing, preserves ≤, and BG e ≤ G for G ∈ W.
(ii) h is increasing, preserves e ≤, and G e ≤ hG for G ∈ W.
It follows from the previous items, that the partial order in Word is as depicted in ( ). We use this order to prove the rest of the theorem by establishing the following.
(1) Each of the 6 words in ( ) is idempotent; that is, w · w = w or w 2 = w.
(2) Word ⊆ the set of the six words in ( ).
(3) Each word in ( ) is distinct. Note, any statement w(x) = · · · , means w(h) = · · · for all h. When convenient we replace the x with h in the following: (1) is a result of the following arithmetic verifications.
(i) B 2 = B and h 2 = h. This is obvious.  (2) Consider a general word w(x) = B or x. Then, w(x) = y n ·y n−1 · · · y 2 ·y 1 , with at least two different terms (that is, both B and x appear), with B's and x's alternating. Replacing x by h and utilizing (1) yields that there exists a 1 ≤ p ∈ N such that (by (ii) of (1)).
Recall w 1 (x) < w 2 (x) means that w 1 (x) ≤ w 2 (x) and there exists an h ∈ hoW for which w 1 (h) < w 2 (h); that is, there exists a G ∈ W such that w 1 (h)G < w 2 (h)G. For the various assertions above only two hull classes are required (probably as F(2) is generated by two elements). Any two of disparate character will suffice (as commented parenthetically below with reference to terminology of section 3). We choose c 3 and e, respectively, the maximum essential reflection and maximum hull operator. Per our observation above, we calculate for every w, w(c 3 ) and w(e) and indicate appropriate G's in W; recall from Example 1.2, It follows from the above calculations along with the appropriate choices for G that: B < BxB, for x = c 3 or x = e (in fact, B = BxB is known only for x = Id or loc). BxB < Bx, for x = c 3 (or any x = h, which is P B op ). BxB < xB, for x = e (or any x = h, which is not P B). Bx < xBx for x = e (or any x = h, which is not P B). xB < xBx for x = c 3 (or any x = h, which is not P B op ). xBx < x for x = c 3 (or any x = h, which is P B). To see xB = Bx, observe if h = c 3 (or any h P B but not P B op ), then hB < Bh. On the other hand, if h = e (or any h antiP B), then Bh < hB.

Equations in Word
An equation is an expression E : and is denoted by h |= E. Since there are 6 words, there are 6 × 6 = 36 possible equations. However, since w 1 = w 2 is the same as w 2 = w 1 , there are only 18 to consider. Moreover, since any h is increasing, it follows that no h satisfies x = B · · · . Consequently, this eliminates x = B, x = BxB and x = Bx and leaves only 15 for consideration. Throughout the phrase "E is an equation" refers to one of these 15.
where the "M " stands for mysterious.
The symbols P B etc. will also be used for the class {h | h is P B} etc. In addition, we note, CB = P B P B op , P B M + = ∅, P B op M + = antiP B.
The facts inserted into Definitions 3.1 are easily shown using repeatedly the identity w 2 = w for every w (item (1) in the proof of Theorem 2.2). Some of the names and reasons for existence, etc. appear in our various papers in the references, in particular [11].
We now present the chart of equations, with explanation to follow. Here N signifies the equation is never satisfied (e.g., B = x), T signifies a tautology that is, w = w. Below the diagonal is the reflection of above and the diagonal is w 1 = w 2 "equals" w 2 = w 1 . The arrows signify implications (that is, inclusion of class) and we do not know if any or all arrows in the B-row reverse. The only B-row examples are Id and loc, which are alB. The ? in the B-row remain un-named and we do not know if antiP B implies M + . All other arrows do not reverse. We note that there is the set of "basic" equations B = {M − , P B, P B op , M + }: each other equation is a conjunction of these. This and the implications can be easily shown (some previously asserted in 3.1).
Recall that E refers to the 15 viable equations.
(1) For every equation E, there are p, n ∈ hoW for which p |= E, n |= E.
(2) There does not exist an h such that h |= E for every E.
Proof. (1) Since by Theorem 2.2 the words are all distinct, it follows that for every E there exists an n ∈ hoW such that n |= E. p = Id or loc is alB, thus p satisfies E for all E = antiP B or M + . p = e, the essential completion, satisfies E = antiP B and M + .

Quotients; a partition of hoW
Especially in this section a picture is worth many words. Here is Word with the vertices relabelled to reduce notation.
Word 4.1: Proposition 4.3. Fix h ∈ hoW. Q(h) is a lattice and σ h is a lattice homomorphism. σ h obeys the following laws: Proof. By definition, Q(h) is partially ordered and σ h preserves order. The facts that Q(h) is a lattice and σ h a lattice homomorphism reduce to the issues that the following two equations It is easy to see that if Bh = hB in Q(h), then E1, and E2 hold. In the case, Bh = hB in Q(h), then Q(h) is the 3-chain (see Figure 6 below), and again we have equality.   We refine the process further towards our partition of hoW with notation and pictures, which we trust are obvious by now.  (b) (i) Id and loc ∈ alB (and are the only ones we know).
(ii) c, the Dedekind completion of the divisible hull, has c ∈ CB \ alB.
We do not know if every σ h is one of the above four. Those four are all chains. We don't know if every Q(h) is a chain (see Sections 5 and 6).
We have the Venn Diagram: Here, hoW is the disc.  (ii) alB contains at least two elements, Id and loc. We know no more. See section 6 below.
(iii) P B \ CB contains many, not all, essential reflections, and is at least infinite (revealed from the parameterizations of these in [20]). We are sure more can be said but depart the topic for now.
Some further subdivision of the regions can be found in [9] (especially of P B), and [10] and [11] (especially of CB). The following says that these are the same question, even "locally at h." We do not know the answers, and shall pick apart the proof in the hopes of informing the situation. (1) σ h (λ) and σ h (ρ) are comparable (that is, Bh ≤ hB or hB ≤ Bh).

P B and P B op
The following triviality shows Theorem 5.1 (1) ⇐⇒ (3). Proof. Since f preserves order, we have: where A denotes f (λ) or f (ρ). The assertion is obvious. Lemma 5.3. Suppose Q is partially ordered and Word f −→ Q is a surjection which preserves order and satisfies the laws P B and P B op . The following are equivalent.
(1) Q is a chain.
Proof. The implications ⇐= are clear.
We state the obvious in Corollaries 5.5 and 5.6.
Corollary 5.5. The following are equivalent.
(a) For every h ∈ hoW, either Bh ≤ hB or Bh ≥ hB.
(b) hoW = P B ∪ P B op .
(c) For every h ∈ hoW, Q(h) is a chain.
We don't know if the conditions in Corollary 5.5 hold. Here is "if not." Corollary 5.6. For h ∈ hoW, Q(h) is not a chain if and only if σ h is exactly one of