In
mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite
lattice is
isomorphic to the
congruence lattice of some finite
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
.
Background
A
lattice is called
algebraic if it is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
and
compactly generated In mathematics, compactly generated can refer to:
* Compactly generated group, a topological group which is algebraically generated by one of its compact subsets
*Compactly generated space
In topology, a compactly generated space is a topological s ...
. In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the
congruence lattice of some
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. Thus there is essentially no restriction on the shape of a congruence lattice of an algebra. The finite lattice representation problem asks whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a ''finite'' algebra?
In 1980, Pálfy and Pudlák proved that this problem is equivalent to the problem of deciding whether every finite lattice occurs as an interval in the
subgroup lattice
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their unio ...
of a finite
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. For an overview of the group theoretic approach to the problem, see Pálfy (1993) and Pálfy (2001).
This problem should not be confused with the
congruence lattice problem.
Significance
This is among the oldest unsolved problems in
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular Group (mathematics), groups as ...
.
[Ralph McKenzie. ''Finite forbidden lattices.'' In: Universal algebra and lattice theory
(Puebla, 1982), Lecture Notes in Math., vol. 1004, pp. 176–205. Springer, Berlin
(1983)]
DOI
/ref> Until it is answered, the theory of finite algebras is incomplete since, given a finite algebra, it is unknown whether there are, ''a priori'', any restrictions on the shape of its congruence lattice.
References
Further reading
*
*{{cite journal , last1=Dowling , first1=T A , title=A class of geometric lattices based on finite groups , journal=Journal of Combinatorial Theory, Series B , year=1973 , date=February 1973 , volume=14 , issue=1 , pages=61–86 , doi=10.1016/S0095-8956(73)80007-3, doi-access=free
External links
Finite Congruence Lattice Problem
Algebraic structures
Lattice theory
Mathematical problems
Unsolved problems in mathematics