Coordinatization of lattices by regular rings without unit and Banaschewski functions
A Banaschewski function on a bounded lattice is an antitone self-map of that picks a complement for each element of . We prove a set of results that include the following:
Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism.
Every (not necessarily unital) countable von Neumann regular ring has a map from to the idempotents of such that and for all .
Every sectionally complemented modular lattice with a Banaschewski trace (a weakening of the notion of a Banaschewski function) embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice. This applies, in particular, to any sectionally complemented modular lattice with a countable cofinal subset.
A sectionally complemented modular lattice is coordinatizable, if it is isomorphic to the lattice of all principal right ideals of a von Neumann regular (not necessarily unital) ring . We say that has a large -frame, if it has a homogeneous sequence such that the neutral ideal generated by is . Jónsson proved in 1962 that if has a countable cofinal sequence and a large -frame, then it is coordinatizable. We prove that A sectionally complemented modular lattice with a large -frame is coordinatizable iff it has a Banaschewski trace.
Key words and phrases:Lattice; complemented; sectionally complemented; modular; coordinatizable; frame; neutral; ideal; Banaschewski function; Banaschewski measure; Banaschewski trace; ring; von Neumann regular; idempotent
2000 Mathematics Subject Classification:06C20, 06C05, 03C20, 16E50
Bernhard Banaschewski proved in [Bana] that on every vector space , over an arbitrary division ring, there exists an order-reversing (we say antitone) map that sends any subspace of to a complement of in . Such a function was used in [Bana] for a simple proof of Hahn’s Embedding Theorem that states that every totally ordered abelian group embeds into a generalized lexicographic power of the reals.
By analogy with Banaschewski’s result, we define a Banaschewski function on a bounded lattice as an antitone self-map of that picks a complement for each element of (Definition 3.1). Hence Banaschewski’s above-mentioned result from [Bana] states that the subspace lattice of every vector space has a Banaschewski function. This result is extended to all geometric (not necessarily modular) lattices in Saarimäki and Sorjonen [SaSo].
We prove in Theorem 4.1 that Every countable complemented modular lattice has a Banaschewski function with Boolean range. We also prove (Corollary 4.8) that such a Boolean range is uniquely determined up to isomorphism. In a subsequent paper [BanCoord2], we shall prove that the countability assumption is needed.
Then we extend the notion of a Banaschewski function to non-unital lattices, thus giving the notion of a Banaschewski measure (Definition 5.5) and the more general concept of a Banaschewski trace (Definition 5.1)—first allowing the domain to be a cofinal subset and then replacing the function by an indexed family of elements. It follows from [BanCoord2, Lemma 5.2] that every Banaschewski measure on a cofinal subset is a Banaschewski trace. Banaschewski measures are proved to exist on any countable sectionally complemented modular lattice (Corollary 5.6), and every sectionally complemented modular lattice with a Banaschewski trace embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice (Theorem 5.3). In particular (Corollary 5.4),
Every sectionally complemented modular lattice with a countable cofinal subset embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice.
We finally relate Banaschewski functions to the problem of von Neumann coordinatization. We recall what the latter is about. A ring (associative, not necessarily unital) is von Neumann regular, if for each there exists such that (cf. Fryer and Halperin [FrHa54], Goodearl [Good91]). The set of all principal right ideals of a (not necessarily unital) von Neumann regular ring , that is,
ordered by inclusion, is a sublattice of the lattice of all ideals of ; hence it satisfies the modular law,
(Here denotes the addition of ideals.) Moreover, is sectionally complemented (cf. [FrHa54, Section 3.2]), that is, for all principal right ideals and such that , there exists a principal right ideal such that . A lattice is coordinatizable, if it is isomorphic to for some von Neumann regular ring ; then we say that coordinatizes . In particular, every coordinatizable lattice is sectionally complemented modular. One of the weakest known sufficient conditions, for a sectionally complemented modular lattice, to be coordinatizable, is given by a result obtained by Bjarni Jónsson in 1960, see [Jons60]:
Jónsson’s Coordinatization Theorem.
Every complemented modular lattice that admits a large -frame, or which is Arguesian and that admits a large -frame, is coordinatizable.
We refer to Section 2 for the definition of a large -frame. Jónsson’s result extends von Neumann’s classical Coordinatization Theorem; his proof has been recently substantially simplified by Christian Herrmann [Herr]. On another track, the author proved that there is no first-order axiomatization for the class of all coordinatizable lattices with unit [CXCoord].
We introduce a ring-theoretical analogue of Banaschewski functions (Definition 3.4), and we prove that a unital von Neumann regular ring has a Banaschewski function iff the lattice has a Banaschewski function (Lemma 3.5). Interestingly, the definition of a Banaschewski function for a ring does not involve the unit; this makes it possible to prove the following result (cf. Corollary 4.6):
For every countable (not necessarily unital) von Neumann regular ring , there exists a map from to the idempotents of such that and for all .
Finally, we relate coordinatizability of a lattice and existence of Banaschewski traces on . Our main result in that direction is that A sectionally complemented modular lattice that admits a large -frame, or which is Arguesian and that admits a large -frame, is coordinatizable iff it has a Banaschewski trace (Theorem LABEL:T:CharCoord4Fr).
2. Basic concepts
By “countable” we will always mean “at most countable”. We shall denote by the set of all non-negative integers.
Let be a partially ordered set. We denote by (resp., ) the least element (resp. largest element) of when they exist, also called zero (resp., unit) of , and we simply write (resp., ) in case is understood. Furthermore, we set . We set
for any subsets and of , and we set , , for any . We say that is a lower subset (resp., upper subset) of , if (resp., ). We say that is upward directed, if every pair of elements of is contained in for some . We say that is cofinal in , if . An ideal of is a nonempty, upward directed, lower subset of . We set
For partially ordered sets and , a map is isotone (resp., antitone), if implies that (resp., ), for all .
We refer to Birkhoff [Birk94] or Grätzer [GLT2] for basic notions of lattice theory. We recall here a sample of needed notation, terminology, and results. A family of elements in a lattice with zero is independent, if the equality
holds for all finite subsets and of . In case is modular and for a non-negative integer , this amounts to verifying that the equality holds for each . We denote by the operation of finite independent sum in ; hence means that is finite, is independent, and . If is modular, then is both commutative and associative in the strongest possible sense for a partial operation, see [Maed58, Section II.1].
A lattice with zero is sectionally complemented, if for all in there exists such that . For elements , let hold, if . We say that is perspective to , in notation , if there exists such that . We say that is complemented, if it has a unit and every element has a complement, that is, an element such that . A bounded modular lattice is complemented if and only if it is sectionally complemented.
An ideal of a lattice is neutral, if generates a distributive sublattice of for all ideals and of . In case is sectionally complemented modular, this is equivalent to the statement that every element of perspective to some element of belongs to . In that case, the assignment that to a congruence associates the -block of is an isomorphism from the congruence lattice of onto the lattice of all neutral ideals of .
An independent finite sequence in a lattice with zero is homogeneous, if the elements are pairwise perspective. An element is large, if the neutral ideal generated by is .
A pair , with independent, is a
-frame, if for each with ;
large -frame, if it is an -frame and is large.
The assignment extends canonically to a functor from the category of all regular rings with ring homomorphisms to the category of sectionally complemented modular lattices with -lattice homomorphisms (cf. Micol [Micol] for details). This functor preserves direct limits.
Denote by the set of all idempotent elements in a ring . For idempotents and in a ring , let hold, if ; equivalently, .
We shall need the following folklore lemma.
Let and be right ideals in a ring and let be an idempotent element of . If , then there exists a unique pair such that . Furthermore, both and are idempotent, , , and .
3. Banaschewski functions on lattices and rings
Let be a subset in a bounded lattice . A partial Banaschewski function on in is an antitone map such that for each . In case , we say that is a Banaschewski function on .
Trivially, every bounded lattice with a Banaschewski function is complemented. The following example shows that the converse does not hold as a rule.
The finite lattice diagrammed on Figure 1 is complemented. However, does not have any Banaschewski function, because is the unique complement of , is the unique complement of , , while .
Although most lattices involved in the present paper will be modular, it is noteworthy to observe that Banaschewski functions may also be of interest in the ‘orthogonal’ case of meet-semidistributive lattices. By definition, a lattice is meet-semidistributive, if implies that , for all . The following result has been pointed to the author by Luigi Santocanale.
Let be finite lattice. Consider the following conditions:
the set of all atoms of joins to the largest element of ;
has a Banaschewski function;
Then (ii) implies (iii) implies (i). Furthermore, if is meet-semidistributive, then (i), (ii), and (iii) are equivalent.
Denote by the set of all atoms of .
(i)(ii) in case is meet-semidistributive. Set
for each . For and , if , then , thus, as is an atom, , thus, by the definition of , , a contradiction. Thus for each , and thus, by assumption, . Furthermore, it follows from the meet-semidistributivity of that , for each . As is obviously antitone, is a Banaschewski function on .
(ii)(iii) is trivial.
(iii)(i). Set . As is complemented, there exists such that . If is nonzero, then there exists an atom below , thus , a contradiction. Hence , and so . ∎
The conditions (i)–(iii) of Proposition 3.3 are not uncommon. They are, for example, satisfied for the permutohedron on a given finite number of letters. It follows that they are also satisfied for the associahedron (Tamari lattice), which is a quotient of the permutohedron.
We shall now introduce a ring-theoretical analogue of the definition of a Banaschewski function.
Let be a subset in a ring . A partial Banaschewski function on in is a mapping such that
for each .
implies that , for all .
In case we say that is a Banaschewski function on .
In the context of Definition 3.4, we put
Banaschewski functions in rings and in lattices are related by the following result.
Let be a unital von Neumann regular ring and let . Then the following are equivalent:
There exists a partial Banaschewski function on in .
There exists a partial Banaschewski function on in .
(i)(ii). Let be a partial Banaschewski function. For each , as it follows from Lemma 2.1 that the unique element such that is idempotent and satisfies both relations and . Let such that . From and the idempotence of it follows that . From
together with the idempotence of we get , and thus . Therefore, .
(ii)(i). Let be a partial Banaschewski function. As
there exists a unique map such that
and is antitone. Furthermore, for each , from the idempotence of it follows that , that is, . Therefore, is a partial Banaschewski function on in . ∎
4. Banaschewski functions on countable complemented modular lattices
A large part of the present section will be devoted to proving the following result.
Every countable complemented modular lattice has a Banaschewski function with Boolean range.
Let be a complemented modular lattice. We denote by the set of all finite sequences , where , of elements of such that . We set , and, further, for each (with ). Furthermore, for each we set
The following statements hold, for each and each :
(i). As , it suffices to prove that for each . We argue by induction on ; the induction hypothesis is that. If then, by the induction hypothesis, as well, while if , that is, , then .
(ii). For each , from it follows a fortiori that . Therefore, writing with , we obtain, by using the modularity of , that the finite sequence is independent in . In particular,
(iii) follows immediately from the containment . ∎
Let and let . If , then .
From the inequality it follows that . The conclusion follows immediately from the definition of . ∎
For and isotone and surjective, let hold, if
(observe that the join in (4.1) is necessarily independent). We say that refines , if there exists such that . Then we denote by (resp., ) the least (resp., largest) element of , for each . As is isotone and surjective, and .
Say that an element decides an element , if . By Lemma 4.2(iii), it follows that .
Let , let , and let . Then the following statements hold:
and , for each .
if refines and decides , then decides and .
(i) follows easily from (4.1).
(ii). Let and set . From together with (i) it follows that , that is, .
(iii). Let , so belongs to , that is, . As is modular and by (4.1), this means that the finite sequence
is independent, thus, as ,
that is, by (4.1), , which means that .
(iv). For each , it follows from (i) that and from (ii) that , thus . As this holds for each , we obtain that .
(v). Let . It follows from (iii) that , thus, by (4.1), . This holds for each , thus .
(vi). As , we obtain, by using (ii) and (iii),
so decides . As both and decide , we obtain that and , so the conclusion follows from (iv) and (v). ∎
For each and each , there exists such that refines and decides .
Set . For each , we set and we pick such that . It is obvious that the finite sequence belongs to and refines .
It remains to verify that decides . So let . If for some , then . Suppose that for some . As for each while , we get
so . ∎
Proof of Theorem 4.1.
As is countable, we can write and denote by the least non-negative integer such that , for each . It follows from Lemmas 4.4(vi) and 4.5 that there exists a sequence of elements of such that decides all elements , …, and refines , for each . We set , for each . Observe that, by Lemma 4.4(vi), for each integer . From Lemma 4.2 it follows that . Finally, from Lemma 4.3 it follows that the map is antitone, so it is a Banaschewski function on .
Furthermore, (the underlying set of) each is independent with join , thus it generates a Boolean sublattice of with the same bounds as . As refines , contains . As the range of each is contained in , the range of is contained in the Boolean sublattice of . For each , is a complement of in , thus it is the unique complement of in —denote it by . As , it follows that the range of is exactly . ∎
For von Neumann regular rings we get the following corollary.
Every countable von Neumann regular ring has a Banaschewski function.
We emphasize that we do not require the ring be unital in Corollary 4.6.
Let be a countable von Neumann regular ring. By Fuchs and Halperin [FuHa64], embeds as a two-sided ideal into some unital von Neumann regular ring . Starting with and closing under the ring operations and a given operation of quasi-inversion on , we obtain a countable von Neumann regular subring of containing ; hence we may assume that is countable. It follows from Theorem 4.1 that has a Banaschewski function. By Lemma 3.5, it follows that has a Banaschewski function, say . For each , as and is a right ideal of , belongs to . Furthermore, there exists such that , thus, as is idempotent, . As is a two-sided ideal of , belongs to , and thus belongs to . As , it follows that . It follows that the restriction of from to is a Banaschewski function on . ∎
Say that a Banaschewski function on a lattice is Boolean, if its range is a Boolean sublattice of . In case is the subspace lattice of a vector space , the range of a Boolean Banaschewski function on may be chosen as the set of all spans of all subsets of a given basis of . In particular, is far from being unique.
However, we shall now prove that if is a countable complemented modular lattice, then is unique up to isomorphism. For a Boolean algebra and a commutative monoid , a V-measure (cf. Dobbertin [Dobb83]) from to is a map such that if and only if , for all disjoint , and if , then there are such that , , and .
Denote by the canonical map from to its dimension monoid , see page 259 and Chapter 9 in Wehrung [WDim].
Let be a Banaschewski function with Boolean range on a complemented modular lattice . Then the restriction of from to is a V-measure on .
It is obvious that if and only if , for each , and that whenever and are disjoint elements in (for they are also disjoint in ). Now let and let such that . It follows from [WDim, Corollary 9.6] that there are such that , , and .
Put . As both and belong to , the element also belongs to . Furthermore, , and
so and so and are perspective. In particular, .
Likewise, there exists such that , so . ∎
For Boolean algebras and , a subset of is an additive V-relation, if , if and only if , if and only if there exists a decomposition with and , and symmetrically with and interchanged. Vaught’s isomorphism Theorem (cf. [Pier, Theorem 1.1.3]) implies that any additive V-relation between countable Boolean algebras and contains the graph of some isomorphism from onto .
In particular, if and are Boolean algebras, then, for any V-measures and such that , the binary relation
is an additive V-relation between and . Therefore, if both and are countable, then, by Vaught’s Theorem, there exists an isomorphism such that .
By the above paragraph, we obtain
Let be a countable complemented modular lattice. Then for a Boolean Banaschewski function on with range , the pair is unique up to isomorphism. In particular, is unique up to isomorphism.
5. Banaschewski measures and Banaschewski traces
A Banaschewski trace on a lattice with zero is a family of elements in , where is an upward directed partially ordered set with zero, such that
for all in ;
is cofinal in .
We say that the Banaschewski trace above is normal, if and implies that , for all .
It is trivial that every bounded lattice has a normal Banaschewski trace (if take and ; if is bounded nontrivial take and while ), so this notion is interesting only for unbounded lattices.
It is obvious that every sectionally complemented modular lattice embeds into a reduced product of its principal ideals, thus into a complemented modular lattice. Our first application of Banaschewski traces, namely Theorem 5.3, deals with the question whether such an embedding can be taken with ideal range. We will use the following well-known lemma.
Lemma 5.2 (Folklore).
Let , , be elements in a modular lattice . If, then and .
It is not hard to verify that the conclusion of Lemma 5.2 can be strengthened by stating that the sublattice of generated by is distributive.
We start by computing, using the modularity of and the assumption,
It follows that
It follows, by using again the modularity of , that
Every sectionally complemented modular lattice with a Banaschewski trace embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice.
Let be a Banaschewski trace in a sectionally complemented modular lattice . The conclusion of the theorem for is trivial in case has a unit, so suppose that has no unit.
We denote by the filter on generated by all principal upper subsets , for , and we denote by the reduced product of the family modulo . For any and any family in , we shall denote by the equivalence class modulo of the family defined by
In particular, for each , the subset contains a principal filter of , thus we can define a map by the rule
Furthermore, for each , define a map by the rule
Consider the following subset of .
The following claim shows that the union on the right hand side of (5.1) is directed.
implies that , for all .
Proof of Claim..
For all ,
|∎ Claim 1.|
Now it is obvious that is a -lattice embedding from into , while is a join-homomorphism, for each . Furthermore, , for all and all . In particular, by Claim 1, the subset defined in (5.1) is a -subsemilattice of .
Let and let . Then both equalities and hold. In particular, is a lattice homomorphism from to .
Proof of Claim..
Let . From and it follows that . By Lemma 5.2, we obtain the following equations:
Therefore, by evaluating the equivalence class modulo of both sides of each of the equalities above as , we obtain the desired conclusion. ∎ Claim 2.
In particular, from Claims 1 and 2 it follows that is a meet-subsemilattice of . Therefore, is a -sublattice of . As is a reduced product of sublattices of , it belongs to the same quasivariety as ; hence so does .
Furthermore, for all and all such that , if , then, by Claim 2,
thus belongs to . Therefore, is an ideal of .
Now we verify that is a complemented modular lattice. It has a unit, namely . Let and let such that . As is sectionally complemented, there exists such that . Hence
while, by Claim 2,
Therefore, . By symmetry between and , we also obtain . Therefore, is complemented.
It remains to prove that is a neutral ideal of . By [Birk94, Theorem III.20], it suffices to prove that contains any element of perspective to some element of . By using Claim 1, it suffices to prove that for any and any , none of the relations and can occur.
If , then , thus there exists such that
In particular, , thus , for each . This contradicts the assumption that has no unit.
The other possibility is . In such a case, , thus, a fortiori, , that is, for all large enough . As has no unit, this is impossible. ∎
Every sectionally complemented modular lattice with a countable cofinal subset has a Banaschewski trace. Hence it embeds, as a neutral ideal and within the same quasivariety, into some complemented modular lattice.
Let be a sectionally complemented modular lattice with an increasing cofinal sequence . We may assume that . Pick such that , for each , and set , for all non-negative integers . It is straightforward to verify that the family is a Banaschewski trace in . The second part of the statement of Corollary 5.4 follows from Theorem 5.3. ∎
The following definition gives an analogue, for lattices without unit, of Banaschewski functions.
Let be a subset in a lattice with zero. A -valued Banaschewski measure on is a map , , isotone in and antitone in , such that for all in .
Our subsequent paper [BanCoord2] will make a heavy use of Banaschewski measures.
Every countable sectionally complemented modular lattice has a Banaschewski measure on .