27 research outputs found
Algebra and the Complexity of Digraph CSPs: a Survey
We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems
Affine hom-complexes
For two general polytopal complexes the set of face-wise affine maps between
them is shown to be a polytopal complex in an algorithmic way. The resulting
algorithm for the affine hom-complex is analyzed in detail. There is also a
natural tensor product of polytopal complexes, which is the left adjoint
functor for Hom. This extends the corresponding facts from single polytopes,
systematic study of which was initiated in [6,12]. Explicit examples of
computations of the resulting structures are included. In the special case of
simplicial complexes, the affine hom-complex is a functorial subcomplex of
Kozlov's combinatorial hom-complex [14], which generalizes Lovasz' well-known
construction [15] for graphs.Comment: final version, to appear in Portugaliae Mathematic
Filtered colimit elimination from Birkhoff's variety theorem
Birkhoff's variety theorem, a fundamental theorem of universal algebra,
asserts that a subclass of a given algebra is definable by equations if and
only if it satisfies specific closure properties. In a generalized version of
this theorem, closure under filtered colimits is required. However, in some
special cases, such as finite-sorted equational theories and ordered algebraic
theories, the theorem holds without assuming closure under filtered colimits.
We call this phenomenon "filtered colimit elimination," and study a sufficient
condition for it. We show that if a locally finitely presentable category
satisfies a noetherian-like condition, then filtered colimit
elimination holds in the generalized Birkhoff's theorem for algebras relative
to .Comment: 22 page
Algebrák és kísérőstruktúráik = Algebras and their related structures
Bebizonyítottuk, hogy - a várakozásokkal ellentétben - véges algebrákra eldönthető, hogy van-e n-változós többségi kifejezésfüggvényük valamely n-re, és hogy véges csoportok izomorfiája nem következik a köbeik részcsoporthálójának izomorfiájából. Megmutattuk, hogy minden végesen generált, relatív kongruencia-metszet-féligdisztributív kvázivarietás rendelkezik véges kváziazonosság-bázissal, s hogy lokálisan véges varietásban a véges algebrák kompatibilis részbenrendezett halmazaira (topológiáira) kirótt különböző feltételek ekvivalensek azzal, hogy a varietás típushalmazában az öt Hobby-McKenzie-féle típus közül bizonyosak nem szerepelnek. Beláttuk, hogy minden E-tömör lokálisan inverz félcsoport beágyazható teljesen egyszerű félcsoport inverz félcsoporttal vett lambda-szemidirekt szorzatába. Bebizonyítottuk, hogy az a probléma, hogy létezik-e minden véges inverz monoidnak véges F-inverz fedője, ekvivalens a véges relatívan szabad csoportok Cayley-gráfjainak bizonyos tulajdonságával, s ennek segítségével a probléma visszavezethető véges inverz monoidoknak egy viszonylag szűk halmazára. Optimális Malcev-feltételeket konstruáltunk a modularitásnál erősebb hálóazonosságokra. Leírást adtunk nagy szimmetriával rendelkező klónokra, bizonyos centralizátor klónokra, továbbá olyan klónokra, amelyek unér része (maximális) inverz monoid. | We proved that - contrary to expectation - it is decidable for a finite algebra whether it has a near unanimity term operation, and that the ismorphism of finite groups does not follow from the isomorphism of the subgroup lattices of their direct cubes. We showed that every finitely generated, relatively congruence meet-semidistributive quasivariety has a finite basis of quasi-identities, and that in a locally finite variety certain conditions imposed on the compatible partial orders (topologies) of the finite algebras are equivalent to omissions of some of the five Hobby-McKenzie types from the type set of the variety. We proved that every E-solid locally inverse semigroup can be embedded in a lambda-semidirect product of a completely simple semigroup by an inverse semigroup. We showed that the problem whether every finite inverse monoid has a finite F-inverse cover is equivalent to a certain property of the Cayley graphs of finite, relatively free groups, and therefore the problem can be reduced to a relatively small class of finite inverse monoids. We constructed optimal Mal'tsev conditions for lattice identities that are stronger than modularity. We described clones with a high degree of symmetry, certain centralizer clones, and clones whose unary part is a (maximal) inverse monoid
Series-Parallel Posets and Polymorphisms
We examine various aspects of the poset retraction problem for series-parallel posets. In particular we show that the poset retraction problem for series-parallel posets that are already solvable in polynomial time are actually also solvable in nondeterministic logarithmic space (assuming P 6= NP). We do this by showing that these series-parallel posets when expanded by constants have bounded path duality. We also give a recipe for constructing members of this special class of series-parallel poset analogous to the construction of all series-parallel posets. Piecing together results from [5],[15],[14] and [12] one can deduce that if a relational structure expanded by constants has bounded path duality then it admits SD-join operations. We directly prove the existence of SD-join operations on members of this class by providing an algorithm which constructs them. Moreover, we obtain a polynomial upper bound to the length of the sequence of these operations. This also proves that for this class of series-parallel posets, having bounded path duality when expanded by constants is equivalent to admitting SD-join operations. This equivalence is not yet known to be true for general relational structures; only the forward direction is proven. However the reverse direction is known to be true for structures that admit NU operations. Zádori has classified in [26] the class of series-parallel posets admitting an NU operation and has shown that every such poset actually admits a 5-ary NU operation. We give a recipe for constructing series-parallel posets of this class analogous to the one mentioned before. Then we show an alternative proof for Zádori's result
Cubical Approximation for Directed Topology II
The paper establishes an equivalence between localizations of (diagrams of)
cubical sets and (diagrams of) directed topological spaces by those maps
defining (natural) cubical homotopy equivalences after application of the
directed singular functor and a directed analogue of fibrant replacement. This
equivalence both lifts and extends an equivalence between classical homotopy
categories of cubical sets and topological spaces. Some simple applications
include combinatorial descriptions and subsequent calculations of directed
homotopy monoids and directed singular 1-cohomology monoids. Another
application is a characterization of isomorphisms between small categories up
to zig-zags of natural transformations as directed homotopy equivalences
between directed classifying spaces. Cubical sets throughout the paper are
taken to mean presheaves over the minimal symmetric monoidal variant of the
cube category. Along the way, the paper characterizes morphisms in this variant
as the interval-preserving lattice homomorphisms between finite Boolean lattice
and describes some of the test model structure on presheaves over this variant.Comment: 54 pages, 1 illustration, submissio