In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
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the congruencelatticeproblem asks whether every algebraic distributive lattice is isomorphic to the congruencelattice of some other lattice. The problem...
finite lattice representation problem, or finite congruencelatticeproblem, asks whether every finite lattice is isomorphic to the congruencelattice of...
rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object...
groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups. Congruence subgroups of...
Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to the congruencelattice of some finite algebra? Goncharov...
COIN-OR Linear Program Solver Communication Linking Protocol Congruencelatticeproblem Constraint Logic Programming Constraint logic programming (Real)...
semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup operation induces a binary operation ∘ on the congruence classes:...
positive integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way....
structural information and can be used for determining, e.g., the congruence relations of the lattice. Triadic concept analysis replaces the binary incidence relation...
structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed...
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed...
Unimodular lattice Fermat's theorem on sums of two squares Proofs of Fermat's theorem on sums of two squares Riemann zeta function Basel problem on ζ(2)...
and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean...
smallest congruence on S such that S/σ is a group, that is, if τ is any other congruence on S with S/τ a group, then σ is contained in τ. The congruence σ is...
This is denoted as z1 ≡ z2 (mod z0). The congruence modulo z0 is an equivalence relation (also called a congruence relation), which defines a partition of...
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula...
that θ is distributive, if it is a join, in the congruencelattice Con S of S, of monomial join-congruences of S. The following definition originates in...
foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms. Congruence and similarity are generalized in...
Digital Calculating Machinery: 141–146. Thomson, W. E. (1958). "A Modified Congruence Method of Generating Pseudo-random Numbers". The Computer Journal. 1 (2):...
Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic group Fuchsian group Modular group Congruence subgroup...
the set of congruence classes or application of the group operation (as applicable) Subtraction stands for subtraction on the set of congruence classes M...
the congruence { ( y y † , ε ) : y ∈ Y } {\displaystyle \{(yy^{\dagger },\varepsilon ):y\in Y\}} , which is sometimes called the Dyck congruence—in a...
ISBN 978-89-6105-805-6. OCLC 913564239. Sellers, James A. (2013). "An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions". J. Indian Math...
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