arXiv subjects

Friedrich Wehrung

Publications and source records attributed to Friedrich Wehrung.

At least 37 records · Page 2Linked to original sources

arXiv2011

Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.

math.RA↗

arXiv2010

A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame

A sectionally complemented modular lattice L is coordinatizable if it is isomorphic to the lattice L(R) of all principal right ideals of some von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. Jónsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable; whether the cofinal sequence assumption could be dispensed with was left open. We solve this problem by finding a non-coordinatizable sectionally complemented modular lattice L with a large 4-frame; it has cardinality aleph one. Furthermore, L is an ideal in a (necessarily coordinatizable) complemented modular lattice with a spanning 5-frame. Our proof uses Banaschewski functions. A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. In an earlier paper, we proved that every countable complemented modular lattice has a Banaschewski function. We prove that there exists a unit-regular ring R of cardinality aleph one and index of nilpotence 3 such that L(R) has no Banaschewski function.

math.RA↗

arXiv2010

An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowski's Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first author's result that (l^{+n},n,l)-->n+2, for each infinite cardinal l and each positive integer n, which solves a problem stated in the 1985 monograph of Erdös, Hajnal, Mate, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation (l^{+(n-1)},r,l)-->2^m, where m is the largest integer below (1/2)(1-2^{-r})^{-n/r}, for every infinite cardinal l and all positive integers n and r with r larger than 1 but smaller than n. For example, (\aleph_{210},4,\aleph_0)-->32,768. Other order-dimension estimates yield relations such as (\aleph_{109},4,\aleph_0)--> 257 (using an estimate by Füredi and Kahn) and (\aleph_7,4,\aleph_0)-->10 (using an exact estimate by Dushnik).

math.CO↗

arXiv2009

Large semilattices of breadth three

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $κ$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $κ^{+n}$ and breadth n+1, in which every principal ideal has less than $κ$ elements.

math.GM↗

arXiv2009

Coordinatization of lattices by regular rings without unit and Banaschewski functions

A Banaschewski function on a bounded lattice L is an antitone self-map of L that picks a complement for each element of L. We prove a set of results that include the following: (1) Every countable complemented modular lattice has a Banaschewski function with Boolean range, the latter being unique up to isomorphism. (2) Every (not necessarily unital) von Neumann regular ring R has a map e from R to the idempotents of R such that xR=e(x)R and e(xy)=e(x)e(xy)e(x) for all x,y in R. (3) 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 L is coordinatizable, if it is isomorphic to the lattice L(R) of all principal right ideals of a von Neumann regular (not necessarily unital) ring R. We say that L has a large 4-frame, if it has a homogeneous sequence (a_0,a_1,a_2,a_3) such that the neutral ideal generated by a_0 is L. Jónsson proved in 1962 that if L has a countable cofinal sequence and a large 4-frame, then it is coordinatizable. We prove that a sectionally complemented modular lattice with a large 4-frame is coordinatizable iff it has a Banaschewski trace.

math.RA↗

arXiv2008

Embedding properties of endomorphism semigroups

Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, and by Sub(V) (resp., End(V)) the collection of all subspaces (resp., endomorphisms) of a vector space V. We prove various results that imply the following: (1) If X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2^card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),\cap) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2^card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2^card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, of left ideals). Then End(F) has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras endowed with a notion of independence (in particular independence algebras).

math.RA↗

arXiv2007

Poset representations of distributive semilattices

We prove that for any distributive join-semilattice S, there are a meet-semilattice P with zero and a map f:PxP-->S such that f(x,z)<=f(x,y)vf(y,z) and x<=y implies that f(x,y)=0, for all x,y,z in P, together with the following conditions: (i) f(y,x)=0 implies that x=y, for all x<=y in P. (ii) For all x\leq y in P and all a,b in S, if f(y,x)=avb, then there are a positive integer n and a decomposition x=x_0<=x_1<=...<=x_n=y such that f(x_{i+1},x_i) lies either below a or below b, for all i < n. (iii) The subset {f(x,0)|x\in P} generates the semilattice S. Furthermore, any finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive (v,0,1)-semilattices.

math.RA↗

arXiv2007

A solution to Dilworth's Congruence Lattice Problem

We construct a distributive algebraic lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth, from the forties. The lattice D has compact top element and aleph omega+1 compact elements. Our results extend to all algebras possessing a polynomially definable structure of a join-semilattice with a largest element.

math.RA↗

arXiv2007

Embedding coproducts of partition lattices

We prove that the lattice Eq(X) of all equivalence relations on an infinite set X contains, as a 0,1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that Eq(X) also contains, as a sublattice, the coproduct of 2^{card(X)} copies of itself.

math.RA↗

arXiv2006

Distributive congruence lattices of congruence-permutable algebras

We prove that every distributive algebraic lattice with at most $\aleph\_1$ compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The $\aleph\_1$ bound is optimal, as we find a distributive algebraic lattice $D$ with $\aleph\_2$ compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E. T. Schmidt from 1969. Furthermore, $D$ may be taken as the congruence lattice of the free bounded lattice on $\aleph\_2$ generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued "distances" used by B. Jonsson in his proof of Whitman's embedding Theorem. In particular, the semilattice of compact elements of $D$ is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive join-semilattice with zero is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.

math.GM↗

arXiv2006

Non-extendability of semilattice-valued measures on partially ordered sets

For a distributive join-semilattice S with zero, a S-valued poset measure on a poset P is a map m:PxP->S such that m(x,z) <= m(x,y)vm(y,z), and x <= y implies that m(x,y)=0, for all x,y,z in P. In relation with congruence lattice representation problems, we consider the problem whether such a measure can be extended to a poset measure m*:P*xP*->S, for a larger poset P*, such that for all a,b in S and all x <= y in P*, m*(y,x)=avb implies that there are a positive integer n and a decomposition x=z\_0 <= z\_1 <= ... <= z\_n=y in P* such that either m*(z\_{i+1},z\_i) <= a or m*(z\_{i+1},z\_i) <= b, for all i < n. In this note we prove that this is not possible as a rule, even in case the poset P we start with is a chain and S has size $\aleph\_1$. The proof uses a "monotone refinement property" that holds in S provided S is either a lattice, or countable, or strongly distributive, but fails for our counterexample. This strongly contrasts with the analogue problem for distances on (discrete) sets, which is known to have a positive (and even functorial) solution.

math.GM↗

arXiv2006

Von Neumann coordinatization is not first-order

A lattice L is coordinatizable, if it is isomorphic to the lattice L(R) of principal right ideals of some von Neumann regular ring R. This forces L to be complemented modular. All known sufficient conditions for coordinatizability, due first to J. von Neumann, then to B. Jonsson, are first-order. Nevertheless, we prove that coordinatizability of lattices is not first-order, by finding a non-coordinatizable lattice K with a coordinatizable countable elementary extension L. This solves a 1960 problem of B. Jonsson. We also prove that there is no L\_{infinity, infinity} statement equivalent to coordinatizability. Furthermore, the class of coordinatizable lattices is not closed under countable directed unions; this solves another problem of B. Jonsson from 1962.

math.GM↗

arXiv2006

Semilattices of groups and nonstable K-theory of extended Cuntz limits

We give an elementary characterization of those (abelian) semigroups $M$ that are direct limits of countable sequences of finite direct products of monoids of the form $C\cup\{0\}$ for monogenic groups $C$. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of subgroups of $M$, and it makes it possible to express $M$ as a certain submonoid of a direct product $S\times G$, where $S$ is a distributive semilattice with zero and $G$ is an abelian group. When applied to the monoids $V(A)$ appearing in the nonstable K-theory of C*-algebras, our results yield a full description of $V(A)$ for C*-inductive limits $A$ of finite products of full matrix algebras over either Cuntz algebras $O\_n$, where $2\leq n<\infty$, or corners of $O\_{\infty}$ by projections, thus extending to the case including $O\_{\infty}$ earlier work by the authors together with K.R. Goodearl.

math.OA↗

arXiv2005

Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

A (v,0)-semilattice is ultraboolean, if it is a directed union of finite Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice is a retract of some ultraboolean (v,0)-semilattices. This is established by proving that every finite distributive (v,0)-semilattice is a retract of some finite Boolean (v,0)-semilattice, and this in a functorial way. This result is, in turn, obtained as a particular case of a category-theoretical result that gives sufficient conditions, for a functor $ Pi$, to admit a right inverse. The particular functor $ Pi$ used for the abovementioned result about ultraboolean semilattices has neither a right nor a left adjoint.

math.GM↗

arXiv2005

A $K\_0$-avoiding dimension group with an order-unit of index two

We prove that there exists a dimension group $G$ whose positive cone is not isomorphic to the dimension monoid Dim$L$ of any lattice $L$. The dimension group $G$ has an order-unit, and can be taken of any cardinality greater than or equal to $\aleph\_2$. As to determining the positive cones of dimension groups in the range of the Dim functor, the $\aleph\_2$ bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since $G$ has an order-unit of index two, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to $K\_0$, of dimension groups with order-unit of index 2 by unit-regular rings.

math.GM↗

arXiv2005

Embedding simple commutative monoids into simple refinement monoids

Say that a cone is a commutative monoid in which x+y=0 implies that x=y=0. We show that cones (resp. simple cones) of many kinds order-embed or even embed unitarily into refinement cones (resp. simple refinement cones) of the same kind, satisfying in addition various divisibility conditions. We do this in particular for all cones, or for all separative cones, or for all cancellative cones (positive cones of partially ordered abelian groups). We also settle both the torsion-free case and the unperforated case. Most of our results extend to arbitrary commutative monoids.

math.GM↗

arXiv2005

The M\_{3}[D] construction and n-modularity

In 1968, E. T. Schmidt introduced the M\_3[D] construction, an extension of the ﬁve-element nondistributive lattice M\_3 by a bounded distributive lattice D, deﬁned as the lattice of all triples $(x, y, z) \in D^3$ satisfying $x \mm y = x \mm z = y \mm z$. The lattice M\_3[D] is a modular congruence-preserving extension of D. In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity Un such that U1 is modularity and Un+1 is properly weaker than Un. Let Mn denote the variety deﬁned by Un, the variety of n-modular lattices. If L is n-modular, then M\_3[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, $IdM\_3[L] \cong M\_3[Id L]$. We provide an example of a lattice L such that M\_3[L] is not a lattice. This example also provides a negative solution to a problem of R. W. Quackenbush: Is the tensor product $A\otimes B$ of two lattices A and B with zero always a lattice. We complement this result by generalizing the M\_3[L] construction to an M\_4[L] construction. This yields, in particular, a bounded modular lattice L such that $M\_4 \otimes L$ is not a lattice, thus providing a negative solution to Quackenbush's problem in the variety M of modular lattices. Finally, we sharpen a result of R. P. Dilworth: Every ﬁnite distributive lattice can be represented as the congruence lattice of a ﬁnite 3-modular lattice. We do this by verifying that a construction of G. Grätzer, H. Lakser, and E. T. Schmidt yields a 3-modular lattice.

math.GM↗

arXiv2005

Flat semilattices

Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f: A\to B$ be a (v,0)-embedding. Then the canonical map, $f \otimes \id\_S$, of the tensor product $A \otimes S$ into the tensor product $B \otimes S$ is not necessarily an embedding. The (v,0)-semilattice $S$ is flat, if for every embedding $f : A\to B$, the canonical map $f\otimes\id$ is an embedding. We prove that a (v,0)-semilattice $S$ is flat if and only if it is distributive.

math.GM↗