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Friedrich Wehrung

Publications and source records attributed to Friedrich Wehrung.

At least 55 records · Page 3Linked to original sources
arXiv2005

Finitely presented, coherent, and ultrasimplicial ordered abelian groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, the positive cone G^+ is well-founded as a partially ordered set, and the set of minimal elements of (G^+)-{0} is finite. (ii) Torsion-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Z^n, with a finitely generated submonoid of (Z+)^n as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.

math.GM↗
arXiv2005

Finitely presented and coherent ordered modules and rings

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features: (i) Every subring of Q, and every totally ordered division ring, is po-coherent. (ii) For a partially ordered right module A over a po-coherent poring R, A is po-coherent if and only if A is a finitely presented R-module and A^+ is a finitely generated R^+-semimodule. (iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent. (iv) Every finitely presented abelian lattice-ordered group is po-coherent.

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arXiv2005

Tensor products of semilattices with zero, revisited

Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: $A \otimes B$ is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: $Conc A\otimes Conc B \cong Conc(A \otimes B)$. This generalizes from finite lattices to arbitrary lattices the main result of a joint paper by the first author, H. Lakser, and R. W. Quackenbush.

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arXiv2005

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, $\Dim$, from L x L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Conc L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to Z^+ or to R^+. (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of Z^+ $\cup$ {$\infty$}. (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid. (4) If L is a simple geometric lattice, then DimL is isomorphic to Z^+, if L is modular, and to the two-element semilattice, otherwise. (5) If L is a countably meet-continuous complemented modular lattice, then both Dim L and the dimension function $\Dim$ satisfy (countable) completeness properties. If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M\_2(R), then Dim L is isomorphic to the monoid V(R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.

math.GM↗
arXiv2005

A uniform refinement property for congruence lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlak, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice $A$ with $\aleph\_2$ compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice $A$ cannot be obtained using the Pudlak, Tischendorf, Tuma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, which is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

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arXiv2005

Congruence lattices of free lattices in non-distributive varieties

We prove that for any free lattice F with at least $\aleph\_2$ generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F. This solves a question formulated by Grätzer and Schmidt in 1962. This yields in turn further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring.

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arXiv2005

Norm-closed intervals of norm-complete ordered abelian groups

Let $(G,u)$ be an archimedean norm-complete dimension group with order unit. Continuing a previous paper, we study intervals (i.e., nonempty upward directed lower subsets) of $G$ which are closed with respect to the canonical norm of $(G,u)$. In particular, we establish a canonical one-to-one correspondence between closed intervals of $G$ and certain affine lower semicontinuous functions on the state space of $(G,u)$, which allows us to solve several problems of K. R. Goodearl about inserting affine continuous functions between convex upper semicontinuous and concave lower semicontinuous functions. This yields in turn new results about analogues of multiplier groups for norm-closed intervals.

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arXiv2005

From join-irreducibles to dimension theory for lattices with chain conditions

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D\_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative defined by generators D(p), for p in J(L), and relations D(p) +D(q) = D(q), for all p, q in J(L) such that p D\_L q . As a consequence, we obtain the following results: Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the quasi-identity 2x=x implies x=0. Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A $\bp$ B f A and B is join-semidistributive, and Dim(A $\bp$ B) is isomorphic to $Dim A \otimes Dim B$, where $\otimes$ denotes the tensor product of commutative monoids.

math.GM↗
arXiv2005

Simultaneous representations of semilattices by lattices with permutable congruences

The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilatticeS is isomorphic to the semilattice Conc L of compact congruences of a lattice L. While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that $|S|\leq\aleph\_1$. Furthermore, one can then take L with permutable congruences. This contrasts with the case where $|S| \geq\aleph\_2$, where there are counterexamples S for which L cannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well. We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Conc functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square 2^2 can be lifted with respect to the Conc functor. We prove that the latter results cannot be extended to the cube, 2^3. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Conc functor, by lattices with permutable congruences. We also extend many of our results to lattices with almost permutable congruences, that is, $\ga\jj\gb=\ga\gb\uu\gb\ga$, for all congruences a and b. We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Conc functor on finite Boolean semilattices.

math.GM↗
arXiv2005

A new lattice construction: the box product

In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisfies a very natural condition (of being capped) implying that $A\otimes B$ is a lattice. In general, $A \otimes B$ is not a lattice; for instance, we proved that $M\_3\otimes F(3)$ is not a lattice. In this paper, we introduce a new lattice construction, the box product for arbitrary lattices. The tensor product construction for complete lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the tensor product construction of A. Fraser in 1978 for semilattices bear some formal resemblance to the new construction. For lattices A and B, while their tensor product $A\otimes B$ (as semilattices) is not always a lattice, the box product, $A \bp B$, is always a lattice. Furthermore, the box product and some of its ideals behave like an improved tensor product. For example, if A and B are lattices with unit, then the isomorphism $Conc(A $\bp$ B) \cong Conc A \otimes Conc B$ holds. There are analogous results for lattices A and B with zero and for a bounded lattice A and an arbitrary lattice B. A join-semilattice S with zero is called {0}-representable, if there exists a lattice L with zero such that $Conc L \cong S$. The above isomorphism results yield the following consequence: The tensor product of two {0}-representable semilattices is {0}-representable.

math.GM↗
arXiv2005

Tensor products and transferability of semilattices

In general, the tensor product, $A\otimes B$, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If $A \otimes B$ is a capped tensor product, then $A \otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that $A \otimes B$ is a capped tensor product, for every lattice B with zero; we shall call such lattices amenable. The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill [5] defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that a finite lattice A is amenable iff it is sharply transferable as a join-semilattice. For a general lattice A with zero, we obtain the result: A is amenable iff A is locally finite and every finite sublattice of A is transferable as a join-semilattice. This yields, for example, that a finite lattice A is amenable iff $A\otimes F(3)$ is a lattice iff A satisfies (T), with respect to $\jj$. In particular, $M3 \otimes F(3)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

math.GM↗
arXiv2005

Embedding finite lattices into finite biatomic lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, --The class of all finite lattices; --The class of all finite lower bounded lattices (solved by the first author's earlier work). --The class of all finite join-semidistributive lattices (this problem was, until now, open). We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.

math.GM↗
arXiv2005

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence lattice of L. It is well-known that the inequality je(L) $\ge$ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | Con L isomorphic to D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) $\le$ (2/3)| J(D)|, for any finite distributive lattice D; the constant 2/3 is best possible. A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.

math.GM↗
arXiv2005

Representation of algebraic distributive lattices with $\aleph\_1$ compact elements as ideal lattices of regular rings

We prove the following result: Theorem. Every algebraic distributive lattice D with at most $\aleph\_1$ compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the $\aleph\_1$ bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively, that L has a largest element. This extends several known results of G.M. Bergman, A.P. Huhn, J. T˚uma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma, and G. Grätzer, H. Lakser, and the author.

math.GM↗
arXiv2005

Congruence amalgamation of lattices

J. Tuma proved an interesting "congruence amalgamation" result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: --A.P. Huhn proved that every distributive algebraic lattice $D$ with at most $\aleph\_1$ compact elements can be represented as the congruence lattice of a lattice $L$. We show that $L$ can be constructed as a locally finite relatively complemented lattice with zero. --We find a large class of lattices, the $ω$-congruence-finite lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruence-preserving extension.

math.GM↗
arXiv2005

Solutions to five problems on tensor products of lattices and related matters

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following. Theorem 2. Let A be a lattice with zero. If $A \otimes L$ is a lattice for every lattice L with zero, then A is locally finite and $A \otimes L$ is a capped tensor product for every lattice L with zero. Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that $K \otimes K$ is a capped tensor product. Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.

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