Jiri Tuma, Friedrich Wehrung
We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that every lattice with more than one element has a proper congruence-preserving extension.
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 relati…
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 latt…
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 boun…
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.
Let S be a distributive {∨, 0}-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let $ϕ$: Con K $\to$ S be a {∨, 0}-homomorphism. Then $ϕ$ is, up to isomorphism, of the form Conc f, for a lattice L and a lattice homomorphism f : K $\to$ L. In the statement above, Conc K denotes as usual the {∨, 0}-semilattice of all finitely generated congruences of K. We prove here that this statement characterizes S being a lattice.
The congruence lattices of all algebras defined on a fixed finite set $A$ ordered by inclusion form a finite atomistic lattice $\mathcal E$. We describe the atoms and coatoms. Each meet-irreducible element of $\mathcal E$ being determined by a single unary mapping on $A$, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set $A$. Using these characterizations we deduce several properties of the lattice $\mathcal E$; in particular, we prove that $\mathcal E$ is tolerance-simple whenever $|A|\geq 4$.
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, sect…
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…
We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let $ϕ$: Con K $\to$ D be a {∨, 0}-homomorphism, where Conc K denotes the {∨, 0}-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f : K $\to$ L, and an isomorphism $\ga$: Conc L $\to$ D such that $\ga$ Conc f = $ϕ$. Furthermore, L and f satisfy many additional properties, for example: (i) L is relatively complemented. (ii) L has definable principal congruences. (iii) If the range of $ϕ$ is cofinal in D, then th…
We say that a (∨,0)-semilattice S is conditionally co-Brouwerian, if (1) for all nonempty subsets X and Y of S such that X $\leq$ Y (i.e., x $\leq$ y for all (x, y) $\in$ X x Y), there exists z $\in$ S such that X $\leq$ z $\leq$ Y, and (2) for every subset Z of S and all a, b $\in$ S, if a $\leq$ b ∨ z for all z $\in$ Z, then there exists c $\in$ S such that a $\leq$ b ∨ c and c $\leq$ Z. By restricting this definition to subsets X, Y, and Z of less than $κ$ elements, for an infinite cardinal $κ$, we obtain the definition of a conditionally $κ$- co-Brouweri…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。