On zerodimensionality and the connected component
of locally pseudocompact groups
^{†}^{†}thanks: 2010 Mathematics Subject Classification: Primary 22A05, 54D25, 54H11;
Secondary 22D05, 54D05, 54D30
Abstract
A topological group is locally pseudocompact if it contains a nonempty open set with pseudocompact closure. In this note, we prove that if is a group with the property that every closed subgroup of is locally pseudocompact, then is dense in the component of the completion of , and is zerodimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that may fail to be zerodimensional even for totally minimal pseudocompact groups.
1 Introduction
A Tychonoff space is zerodimensional if it has a base consisting of clopen (openandclosed) sets. With each topological group are associated functorial subgroups related to connectedness properties of , defined as follows (cf. [11, 1.1.1]):

denotes the connected component of the identity;

denotes the quasicomponent of the identity, that is, the intersection of all clopen sets containing the identity;

denotes the intersection of all kernels of continuous homomorphisms from into zerodimensional groups;

denotes the intersection of all open subgroups of .
It is well known that these subgroups are closed and normal (cf. [19, 7.1], [12, 2.2], and [22, 1.32(b)]). Clearly,
Theorem 1.1. ([19, 7.7, 7.8])
Let be a locally compact group. Then is zerodimensional and
The aim of the present paper is to investigate to what extent the condition of local compactness can be relaxed in Theorem 1.1. Although Theorem 1.1 might appear as a result about connectedness, it has far more to do with different degrees of disconnectedness. Recall that a space is hereditarily disconnected if its connected components are singletons, and is totally disconnected if its quasicomponents are singletons. Clearly,
and by Vedenissoff’s classic theorem, both implications are reversible for locally compact (Hausdorff) spaces, that is, the three properties are equivalent for such spaces (cf. [36]).
It is well known that the quotient is hereditarily disconnected for every topological group (cf. [19, 7.3] and [22, 1.32(c)]). Thus, if the implications () and () are reversible for , then is zerodimensional, and so , then Theorem 1.1 holds for . This phenomenon warrants introducing some terminology. . If in addition
Definition 1.2.
A topological group is Vedenissoff if the quotient is zerodimensional; if in addition , then we say that is strongly Vedenissoff.
Our goal is to identify classes of (strongly) Vedenissoff groups, and to find examples of nonVedenissoff groups that have many compactnesslike properties. The latter will demonstrate how close a group must be to being locally compact (or compact) in order to be Vedenissoff. (Not every Vedenissoff group is strongly Vedenissoff. Indeed, is zerodimensional, but has no proper open subgroups, and so . However, thanks to Theorem 2.7(a) below, these two notions coincide in the class of groups that are considered in this paper.)
A Tychonoff space is pseudocompact if every continuous realvalued map on is bounded. A topological group is locally pseudocompact if there is a neighborhood of the identity such that is pseudocompact. (Clearly, every metrizable locally pseudocompact group is locally compact.) We say that is hereditarily [locally] pseudocompact if every closed subgroup of is [locally] pseudocompact. (Note that the adjective hereditary applies only to closed subgroups here, and not to all subgroups. Indeed, by Corollary 2.6 below, if every subgroup of a topological group is locally pseudocompact, then the group is discrete, which is of no interest for the present paper.)
More than fifteen years ago, Dikranjan showed that hereditarily pseudocompact groups are strongly Vedenissoff (cf. [10, 1.2]). We obtain in this paper a theorem that simultaneously generalizes Theorem 1.1, this result of D.D., and provides a positive solution to a problem posed by Comfort and Lukács (cf. [3, 4.17]).
Theorem A.
Let be a hereditarily locally pseudocompact group. Then is zerodimensional and
The next example shows that the condition of hereditarily local pseudocompactness in Theorem A cannot be replaced with (local) pseudocompactness.
Example 1.3.
Comfort and van Mill showed that for every natural number there exists an abelian pseudocompact group such that is totally disconnected, but (cf. [5, 7.7]). In particular, the converse of the implication () may fail for these groups , and they are not Vedenissoff. This shows that pseudocompact groups need not be Vedenissoff.
Although pseudocompactness alone is too weak a property to imply that the group is Vedenissoff, it turns out that it is sufficient in the presence of some additional compactnesslike properties. Recall that a (Hausdorff) topological group is minimal if there is no coarser (Hausdorff) group topology (cf. [31] and [16]), and is totally minimal if every (Hausdorff) quotient of is minimal (cf. [13]). Equivalently, is totally minimal if every continuous surjective homomorphism is open.
An unpublished result of Shakhmatov states that the converse of () holds for minimal pseudocompact groups. Specifically, Shakhmatov proved that every pseudocompact totally disconnected group admits a coarser zerodimensional group topology, and thus minimal pseudocompact totally disconnected groups are zerodimensional (cf. [10, 1.6]). We prove a generalization of Shakhmatov’s result:
Theorem B.

Every locally pseudocompact totally disconnected group admits a coarser zerodimensional group topology.

Every minimal, locally pseudocompact, totally disconnected group is zerodimensional, and thus strongly Vedenissoff.
Theorem C.
Let be a totally minimal locally pseudocompact group. Then if and only if is zerodimensional, in which case is strongly Vedenissoff.
More than twenty years ago, Arhangelskiĭ asked whether every totally disconnected topological group admits a coarser zerodimensional group topology. Megrelishvilli answered this question in the negative by constructing a minimal totally disconnected group that is not zerodimensional (cf. [23]). In particular, the converse of the implication () fails for minimal groups. Megrelishvilli’s example also shows that local pseudocompactness cannot be omitted from Theorem B.
Our last result is a negative one, and it is a far reaching extension of the result of Comfort and van Mill cited in Example 1.3. Recall that a group is perfectly (totally) minimal if the product is (totally) minimal for every (totally) minimal group (cf. [32]).
Theorem D.
For every natural number or , there exists an abelian pseudocompact group such that is perfectly totally minimal, hereditarily disconnected, but .
There are many known examples of pseudocompact groups for which the equality fails (cf. [9, Theorem 11], [11, 1.4.10], and [3, 4.7, 5.5]). By Theorem C, one has for each of the groups provided by Theorem D, and thus the are not totally disconnected. This shows that the converse of the implication () may fail for totally minimal pseudocompact groups.
The paper is structured as follows. In §2, we recall some wellknown facts on locally pseudocompact and locally compact groups, their topologies, and their connectedness properties. We devote §3 to the proof of Theorem A, while the proofs of Theorems B and C are presented in §4. Finally, in §5, we prove a general theorem concerning embedding of groups with minimality properties as quasicomponents of pseudocompact groups with the same minimality properties, which yields Theorem D.
2 Preliminaries
All topological groups here are assumed to be Hausdorff, and thus Tychonoff (cf. [19, 8.4] and [22, 1.21]). Except when specifically noted, no algebraic assumptions are imposed on the groups; in particular, our groups are not necessarily abelian. A “neighborhood” of a point means an open set containing the point.
Although, in general, there are a number of useful uniform structures on a topological group that induce its topology, in this note, we adhere to the twosided uniformity and the notions of precompactness and completeness that derive from it (cf. [30], [37], [28], [19, (4.11)(4.15)], and [22, Section 1.3]). A fundamental property of this notion of completeness is that for every topological group , there is a complete topological group (unique up to a topological isomorphism) that contains as a dense topological subgroup; in other words, is a group completion of (cf. [28] and [22, 1.46]).
Theorem 2.1. ([22, 1.49(a), 1.51])

Let be a topological group, and a subgroup. Then .

If is a locally compact group, then is complete, that is, .
A subset of a topological group is precompact if for every neighborhood of the identity, there is a finite such that . (Some authors refer to precompact sets as bounded ones.) A topological group is locally precompact if admits a base of precompact neighborhoods at the identity. Since every pseudocompact subset of a topological group is precompact (cf. [7, 1.11]), locally pseudocompact groups are locally precompact.
Weil showed in 1937 that the completion of a locally precompact group with respect to its left or right uniformity admits the structure of a locally compact group containing as a dense topological subgroup (cf. [37]). This (onesided) Weilcompletion coincides with the Raǐkovcompletion constructed in 1946 (cf. [28]). Therefore, is locally precompact if and only if is locally compact.
Theorem 2.2 below, which summarizes the main results of [6] and [7], provides a characterization of (locally) pseudocompact groups. Recall that a subset of a space is a set of the form with each . The topology on is the topology generated by the subsets of . A subset of is open (respectively, closed, dense) if it is open (respectively, closed, dense) in the topology on .
Theorem 2.2. ([6] and [7])
A topological group is [locally] pseudocompact if and only if is [locally] precompact and dense in , in which case [].
Since the topology of groups plays an important role in the present work, we introduce some notations, and then record a few useful facts. We let denote the set of closed subgroups of the topological group , that is, closed subgroups of that are also subsets of , and we set and
Theorem 2.3. ()
Let be a topological group. Then:
We have already mentioned that the adjective hereditary used in the term hereditarily locally pseudocompact applies only to the closed subgroups of a given group. The next theorem shows that it would be uninteresting to interpret hereditary as applying to all subgroups.
Theorem 2.4.
Let be a locally pseudocompact group. If every countable subgroup of is locally pseudocompact, then is discrete.
Before we proceed to the proof of Theorem 2.4, we formulate a wellknown observation that will be frequently used later on too.
Lemma 2.5.
Let be a topological group, a dense subgroup, and an open subgroup of . Then .

The proof consists of two steps.
Step 1. We show that is locally compact and metrizable. Since is locally precompact, its completion is locally compact, and thus by Theorem 2.3(b), is a base at the identity of the topology on . Pick . Since is locally pseudocompact, by Theorem 2.2, is dense in . By Theorem 2.3(a), the topology is a group topology on , and so Lemma 2.5 is applicable. Consequently, by Lemma 2.5, dense in . is
We claim that is finite. Let be a countable subgroup of . Then, by our assumption, is locally pseudocompact, being a countable subgroup of . However, is also a subgroup of the compact group , and thus, by Theorem 2.1, its completion is compact. Consequently, by Theorem 2.2, is pseudocompact. Hence, is finite, because there are no countably infinite homogeneous pseudocompact spaces (cf. [17, 1.3]). This shows that is finite.
Since is dense in , it follows that is finite, and so has countable pseudocharacter. This implies that is metrizable, because every locally compact space of countable pseudocharacter is first countable (cf. [18, 3.3.4]). Therefore, , because by is dense in .
Step 2. We show that is discrete. Since is locally compact, there is a neighborhood of the identity in such that is compact. Then the open subgroup generated by is compactly generated, and in particular, compact (cf. [19, 5.12, 5.13]). Thus, is separable, because it is metrizable, being a subgroup of . Let be a countable dense subgroup of . By our assumption, is locally pseudocompact, and thus it is locally compact, because is metrizable. So, is a countable locally compact group, and therefore is discrete. In particular, is complete, and is discrete. Hence, is also discrete. ∎
Corollary 2.6.
If every subgroup of a topological group is locally pseudocompact, then is discrete. ∎
Finally, we summarize the relationship between connectedness properties of locally pseudocompact groups and their completions.
3 Proof of Theorem A
Theorem A.
Let be a hereditarily locally pseudocompact group. Then is zerodimensional and
In this section, we present the proof of Theorem A. By Theorem 2.7(a), . We have already noted that if is zerodimensional, then is zerodimensional for every hereditarily locally pseudocompact group . Since every (Hausdorff) quotient of a hereditarily locally pseudocompact group is again hereditarily locally pseudocompact, and the quotient is hereditarily disconnected (cf. [19, 7.3] and [22, 1.32(c)]), it suffices to prove the following statement. . Thus, it suffices to show that for every locally pseudocompact group
Theorem 3.1.
Let be a hereditarily locally pseudocompact, hereditarily disconnected group. Then is hereditarily disconnected, and is zerodimensional.
In the setting of Theorem 3.1, if is hereditarily disconnected, then by Theorem 1.1, is zerodimensional (because it is locally compact), and so is zerodimensional too. Thus, it suffices to show that is hereditarily disconnected whenever is so. We prove the contrapositive of this statement, namely, that if is nontrivial, then is nontrivial too. The proof is broken down into several steps: First, it is shown in Proposition 3.2 that Theorem 3.1 holds in the case where the completion of is a direct product of a zerodimensional compact group and the real line . Then, in Proposition 3.4, it is proven that if contains a nontrivial compact connected subgroup, then is nontrivial. Finally, it is shown that if the component of is nontrivial, but contains no compact connected subgroup, then contains as a closed subgroup.
Proposition 3.2.
Let be a zerodimensional compact group, and a dense hereditarily locally pseudocompact subgroup of . Then one has .
In order to prove Proposition 3.2, we recall a notion and a result that is wellknown to profinite group theorists. A topological group is topologically finitely generated if it contains a dense finitely generated group, that is, there exists a finite subset of such that .
Theorem 3.3. ([29, 2.5.1], [15, 2.1(a)])
If is a topologically finitely generated compact zerodimensional group, then is metrizable.

As is locally pseudocompact, by Theorem 2.2, is dense in . The set set in for every , and so , there is such that . . Thus, for every is a
Let be such that and , and put and . Since is a closed subgroup of , it is a compact topologically finitely generated zerodimensional group, and by Theorem 3.3, is metrizable. Thus, the product is metrizable, and so is metrizable, being a subgroup of . On the other hand, is locally pseudocompact, being a closed subgroup of the hereditarily locally pseudocompact group . Therefore, is locally compact. Hence, by Theorem 2.1, is closed not only in , but also in .
Let denote the second projection, and put . Since is compact and is closed in , is a closed map (cf. [18, 3.1.16]), and thus is closed in and is a closed map too. This implies that is surjective, because contains the dense subgroup of . Consequently, is a quotient map, and is topologically isomorphic to a quotient of . Therefore, is not zerodimensional, and by Theorem 1.1, is nontrivial.
Since
Proposition 3.4.
Let be a hereditarily locally pseudocompact group such that the completion is compact. If contains a nontrivial compact connected subgroup, then is nontrivial.
As one may expect, the proof of Proposition 3.4 relies on Dikranjan’s result for hereditarily pseudocompact groups.
Theorem 3.5. ([10, 1.2, 2.6])
Let be a hereditarily pseudocompact group. Then is zerodimensional, , and is dense in , that is, is strongly Vedenissoff.

Let be a nontrivial connected compact subgroup of . By Theorem 2.3(c), is a base at the identity for the topology on . So, we may pick . Since is a normal subgroup of , the set is a subgroup of , and is compact, because both and are compact. Furthermore, is open in , as it contains the set . By Theorem 2.2, is dense in . By Theorem 2.3(a), the topology is a group topology on , and so Lemma 2.5 is applicable. Consequently, by Lemma 2.5, dense in ; in particular, is dense in , and thus, by Theorem 2.1, . is
We show that is hereditarily pseudocompact. To that end, let be a closed subgroup of . Since is compact, it is closed in , and so is a closed subgroup of . Thus, is a closed subgroup of , and by our assumption, is locally pseudocompact. By Theorem 2.2, this implies that is dense in its completion . The group is compact, being a closed subgroup of , and therefore is not only locally pseudocompact, but also pseudocompact. This shows that is hereditarily pseudocompact. Therefore, by Theorem 3.5, is dense in , and hence
In particular, cannot be trivial, as desired. ∎
One last ingredient of the proof of Theorem 3.1 is a result that is often referred to as Iwasawa’s Theorem (although it also relies on the work of Yamabe).
Theorem 3.6. ([38, Theorem 5’], [21, Theorem 13])
Let be a connected locally compact group. Then there is a compact connected subgroup , and closed subgroups such that each is topologically isomorphic to the additive group , and is homeomorphic to

We prove the contrapositive of the theorem. Let be a hereditarily locally pseudocompact group such that . We show that .
Step 1. As is locally precompact, its completion is locally compact. Let be a neighborhood of the identity in such that is compact. Put , the subgroup generated by , and with if necessary, we may assume that is compact from the outset. . We claim that by replacing
Since contains , it is an open subgroup of , and thus it is also closed (cf. [19, 5.5] and [22, 1.10(c)]). Consequently, is open and closed in , and is also hereditarily locally pseudocompact. By Lemma 2.5, one has 2.1, . As is generated by , it is compactly generated, and in particular, it is compact (cf. [19, 5.12, 5.13]). Since is an open subgroup of , by Theorem 1.1, , and so, by Theorem
Step 2. Since is compact, if contains a nontrivial compact connected subgroup, then by Proposition 3.4, is nontrivial, and we are done. Thus, from now on, we assume that contains no nontrivial compact connected subgroups.
If is a compact subgroup of , then , and by Theorem 1.1, is zerodimensional. Thus, every compact subgroup of is zerodimensional. In particular, contains no nontrivial compact connected subgroups. Therefore, by Theorem 3.6, our assumption yields that there is a closed subgroup of such that (that is, is topologically isomorphic to the additive group ).
Step 3. By Theorem 2.3(c), is a base at the identity for the topology on , and so we may pick . Since is a normal subgroup, acts continuously on by conjugation, and the orbit of is a connected subspace of . By Step 2, is zerodimensional. Thus, the orbit of each is a singleton. Therefore, for every and . In particular, the elements of and commute (elementwise). We note that this argument, concerning the commuting of connected and zerodimensional normal subgroups, is due to K. H. Hofmann (cf. [20]).
Let be a topological isomorphism. The continuous surjection
The subgroup is open in , because it contains the set . By Theorem 2.2, is dense in . By Theorem 2.3(a), the topology is a group topology on , and so Lemma 2.5 is applicable. Consequently, by Lemma 2.5, dense in ; in particular, is dense in , and by Theorem 2.1, . Since is a closed subgroup of , it is hereditarily locally pseudocompact. Thus, satisfies the conditions of Proposition 3.2. Consequently, is
4 Proof of Theorems B and C
Recall that a group topology is linear if it admits a base at the identity consisting of subgroups. Since every open subgroup is also closed (cf. [19, 5.5] and [22, 1.10(c)]), every linear group topology is zerodimensional. We prove a slightly stronger version of Theorem B.
Theorem B.

Every locally pseudocompact totally disconnected group admits a coarser linear group topology.

Every minimal, locally pseudocompact, totally disconnected group has a linear topology, and thus it is strongly Vedenissoff.

(a) Since is totally disconnected, , and since is locally pseudocompact, by Theorem 2.7(a), . Thus, . Therefore, the family of open subgroups in forms a base at the identity for a (Hausdorff) group topology on , and it is obviously coarser than the topology of . Clearly, this topology is linear.
(b) follows from (a) and the definition of minimality. ∎
Theorem C.
Let be a totally minimal locally pseudocompact group. Then if and only if is zerodimensional, in which case is strongly Vedenissoff.
5 Proof of Theorem D
Theorem D.
For every natural number or , there exists an abelian pseudocompact group such that is perfectly totally minimal, hereditarily disconnected, but .
In this section, we prove Theorem D by establishing a general construction that allows one to “realize” minimal abelian groups as quasicomponents of minimal pseudocompact groups. A weaker version of Theorem D, which provides totally minimal pseudocompact groups, was announced in [11, 1.4.2]. The novelty of Theorem D, in addition to its complete proof, is that we obtain perfectly totally minimal pseudocompact groups.
Theorem D.
Let be a precompact abelian group that is contained in a connected compact abelian group . Then there exists a pseudocompact abelian group such that and , and in particular,

is minimal, then may be chosen to be minimal;

is totally minimal, then may be chosen to be totally minimal;

is perfectly minimal, then may be chosen to be perfectly minimal;

is perfectly totally minimal, then may be chosen to be perfectly totally minimal.
Theorem D follows a line of “embedding” results, which state that certain (locally) precompact groups embed into (locally) pseudocompact groups as a particular (e.g., functorial) closed subgroup (cf. [4, 2.1], [34], [5, 7.6], [35], [8, 3.6], and [3, 5.6]). The novelty is that minimality properties of the group are inherited by the group that is constructed. By the celebrated ProdanovStoyanov Theorem, every minimal abelian group is precompact (cf. [26] and [27]), and so the condition that the group is precompact is not restrictive at all.
We first show how Theorem D follows from Theorem D, and then proceed to proving the latter. To that end, we recall a characterization due to Stoyanov for groups that are not only perfectly totally minimal, but their powers have the same property too (cf. [32]). For an abelian topological group , let denote the subgroup of elements in for which there exists a positive integer such that for every sequence of integers, one has in . In other words,
Theorem 5.1. ([32], [14, 6.1.18])
Let be a precompact abelian group. Then is perfectly totally minimal for every cardinal if and only if .
We proceed now to proving Theorem D. The proof has two ingredients: A zerodimensional pseudocompact group with good minimality properties, and a discontinuous homomorphism with kernel . The desired group will be the sum the of graph of and the group formed in the product .
Lemma 5.2.
For every infinite cardinal , there exists a pseudocompact zerodimensional group such that:

is perfectly totally minimal;

.

Let denote the set of prime integers, and for , let denote the group of adic integers. Put . We think of elements of as tuples , where and . We define three subgroups of :

such that only for finitely many primes (or equivalently, ); consists of elements

, where is the product of many copies of (or equivalently, consists of elements such that all but countably many coordinates of are zero);

.
We claim that has the desired properties.
The group is dense in , because it contains , which is clearly dense. Thus, is dense in the compact group , and in particular, by Theorem 5.1, . Therefore, by Theorem 2.2, is pseudocompact. By Theorem 2.7(b), is zerodimensional, because is so.
Since is dense in the compact group , in particular, it is dense, and by Theorem 2.1, . Thus,
In order to prove that , it suffices to show that , as . Let denote the “diagonal” subgroup of , that is, the subgroup generated by such that for every and , and we prove that has at least one zero coordinate. Let . In fact, we show a bit more, namely, that every element in

We consider the next lemma part of the folklore of pseudocompact abelian groups (cf. [2, 3.6, 3.10]), and we provide its proof only for the sake of completeness.
Lemma 5.3. ()
Let and be compact topological groups, and let be a surjective homomorphism such that is dense in . Then the graph of is a dense subgroup of the product , and in particular, is pseudocompact.

Let be a nonempty subset of . Without loss of generality, we may assume that is of the form , where is a set in . Pick . Since is surjective, there is such that . The translate is a nonempty set in , and thus we may pick
A last, auxiliary tool in the proof of Theorem D is the following observation.
Remark 5.4.

Put , and let be the group provided by Lemma 5.2. Since , the quotient contains a free abelian group of rank . As , one may pick a surjective homomorphism . The group is divisible, because it is compact and connected (cf. [19, 24.25]). Thus, can be extended to a surjective homomorphism .
Let denote the composition of with the canonical projection . By Theorem 2.2, is dense in , because is pseudocompact. Thus, is dense in , because . Clearly, is surjective. Therefore, by Lemma 5.3, the graph of is dense in the product .
Put . Since is dense in and contained in , the group is dense too. Thus, by Theorem 2.1, , and by Theorem 2.2, is pseudocompact. As is zerodimensional, , and by Theorem 2.7(a),
We check now that
We turn to minimality properties of . Suppose that . The group always contains the product , but in this case, is dense in , and thus it is dense in . Therefore, by Remark 5.4, inherits all minimality properties of . Since is perfectly totally minimal, the product inherits all minimality properties of . This shows (a)(d). ∎
One wonders whether the condition is necessary for parts (a)(d) of Theorem D. If the resulting group is to be totally minimal, then the answer is positive. Dikranjan showed that if is a minimal pseudocompact abelian group then is dense in if and only if is minimal (cf. [10, 1.7]), in which case is the completion of . This settles the question for (b) and (d). The following remark settles the question for (a) and (c).
Remark 5.5.
We note (without a proof) that the techniques of Theorem D can also be used to construct, for every positive integer or , a perfectly minimal pseudocompact dimensional group such that is not minimal, and hence is not dense in .
6 Concluding remarks
One can also define the intersection of all open normal subgroups of a group , and ask about its relationship with the other four functorial subgroups. If a locally compact group admits a base at the identity consisting of neighborhoods that are invariant under conjugation (that is, is socalled balanced or admits Small Invariant Neighborhoods), which is the case for compact or abelian groups, then . There are, however, many locally compact groups that do not have this property.
Examples 6.1.

The semidirect product acts on the compact group by shifts, is locally compact and zerodimensional, and thus is trivial. However, is the smallest open normal subgroup of , and therefore . , where

For , let denote the (locally compact) field of adic numbers. The discrete multiplicative group of nonzero rationals acts on by multiplication. The semidirect product

In general, let be a locally compact group and a subgroup of such that contains no proper invariant open subgroup, and put is equipped with the discrete topology. Then, by Theorem 1.1, the locally compact group has the property that . and , where
Acknowledgements
This work has emerged from the joint work of one of the authors with W. W. Comfort on locally precompact groups; the authors wish to express their heartfelt gratitude to Wis Comfort for the helpful discussions and correspondence. The authors wish to thank Dragomir Djokovic for the valuable correspondence. The authors are grateful to Karen Kipper for her kind help in proofreading this paper for grammar and punctuation.
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