452 research outputs found
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.Comment: 22 pages. Order, to appea
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 . 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.Comment: 8 pages, Proceedings of AAA 70 -- 70th Workshop on General Algebra,
Vienna University of Technology (May 26--29, 2005), to appea
A survey of recent results on congruence lattices of lattices
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
Presenting dcpos and dcpo algebras
Dcpos can be presented by preorders of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are “stable” for the operations. The resulting dcpo algebra has a natural universal characterization and satisfies all the inequational laws satisfied by the generating algebra. Applications include known “coverage theorems” from locale theory
A Universal Characterization of the Double Powerlocale
This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321.
The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join
semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc
Cevian operations on distributive lattices
We construct a completely normal bounded distributive lattice D in which for
every pair (a, b) of elements, the set {x D | a b x} has a
countable coinitial subset, such that D does not carry any binary operation -
satisfying the identities x y (x-y),(x-y)(y-x) = 0, and x-z
(x-y)(y-z). In particular, D is not a homomorphic image of the
lattice of all finitely generated convex {\ell}-subgroups of any (not
necessarily Abelian) {\ell}-group. It has \lambda\infty\lambda$-elementary equivalence.Comment: 23 pages. v2 removes a redundancy from the definition of a Cevian
operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on
the G side
Sublattices of complete lattices with continuity conditions
Various embedding problems of lattices into complete lattices are solved. We
prove that for any join-semilattice S with the minimal join-cover refinement
property, the ideal lattice IdS of S is both algebraic and dually algebraic.
Furthermore, if there are no infinite D-sequences in J(S), then IdS can be
embedded into a direct product of finite lower bounded lattices. We also find a
system of infinitary identities that characterize sublattices of complete,
lower continuous, and join-semidistributive lattices. These conditions are
satisfied by any (not necessarily finitely generated) lower bounded lattice and
by any locally finite, join-semidistributive lattice. Furthermore, they imply
M. Ern\'e's dual staircase distributivity. On the other hand, we prove that the
subspace lattice of any infinite-dimensional vector space cannot be embedded
into any countably complete, countably upper continuous, and countably lower
continuous lattice. A similar result holds for the lattice of all order-convex
subsets of any infinite chain.Comment: To appear in Algebra Universali
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