STABLE PAIRS

2010 ◽  
Vol 20 (02) ◽  
pp. 241-267 ◽  
Author(s):  
K. HENCKELL
Keyword(s):  
Finite Semigroup ◽  
Membership Problem ◽  
Stable Pair ◽  
Stable Pairs ◽  
Relational Morphism

For a finite semigroup S and pseudovariety V, (Y, T) is a V-stable pair of S iff Y ⊆ S, T ≤ S and for any relational morphism R : S ⇝ V with V ∈ V there exists a v ∈ V such that Y ⊆ R-1(v) and T ≤ R-1( Stab (v)). X ≤ S is stable if it is generated by an [Formula: see text]-chain {ai} with aiaj = ai for j < i. Given a relation R : S ⇝ A ∈ A (where A denotes the pseudovariety of aperiodic semigroups) that computes PlA(S), we construct a new relation R∞ : S ⇝ (A(M))# that computes A-stable pairs. This proves the main result of this paper: (Y, T) is an A-stable pair of S iff T ≤ ∪ X for some stableX ≤ PlA(S) and Y ⊆ Y' for some Y' ∈ PlA(S) with Y'x = Y' for all x ∈ X. As a corollary we get that if V is a local pseudovariety of semigroups, then V * A has decidable membership problem.

scholarly journals The subpower membership problem for semigroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1435-1451 ◽  
Author(s):  
Andrei Bulatov ◽  
Marcin Kozik ◽  
Peter Mayr ◽  
Markus Steindl
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Membership Problem ◽  
Clifford Semigroup ◽  
Direct Power ◽  
Nilpotent Semigroup ◽  
Full Transformation ◽  
Commutative Semigroups ◽  
Np Complete ◽  
A Finite Group

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

scholarly journals The KSBA compactification of the moduli space of $$D_{1,6}$$-polarized Enriques surfaces

2021 ◽  
Author(s):  
Luca Schaffler
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
Blow Up ◽  
Unit Cube ◽  
Finite Group Action ◽  
Stable Pair ◽  
Enriques Surfaces ◽  
Secondary Polytope ◽  
Stable Pairs ◽  
A Finite Group

AbstractWe describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the $${\mathbb {Z}}_2^2$$ Z 2 2 -covers of the blow up of $${\mathbb {P}}^2$$ P 2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

scholarly journals Stable Pair Invariants of Local Calabi–Yau 4-folds

2021 ◽  
Author(s):  
Yalong Cao ◽  
Martijn Kool ◽  
Sergej Monavari
Keyword(s):  
Line Bundles ◽  
Zero Section ◽  
Hilbert Schemes ◽  
Intersection Numbers ◽  
Genus Zero ◽  
Stable Pair ◽  
Local Principle ◽  
Low Degree ◽  
Witten Theory ◽  
Stable Pairs

Abstract In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao–Maulik–Toda conjectures for low-degree curve classes and find connections to Carlsson–Okounkov numbers. Some of our verifications involve genus zero Gopakumar–Vafa type invariants recently determined in the context of the log-local principle by Bousseau–Brini–van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao–Maulik–Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

scholarly journals ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM

2018 ◽  
Vol 106 (1) ◽  
pp. 127-142
Author(s):  
MARKUS STEINDL
Keyword(s):  
Finite Semigroup ◽  
Transformation Semigroup ◽  
Membership Problem ◽  
Direct Power ◽  
Full Transformation ◽  
Rees Matrix Semigroups ◽  
Full Transformation Semigroup ◽  
Np Complete ◽  
Matrix Semigroups

Fix a finite semigroup $S$ and let $a_{1},\ldots ,a_{k},b$ be tuples in a direct power $S^{n}$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_{1},\ldots ,a_{k}$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\geq 2$.

COMPLEXITY PSEUDOVARIETIES ARE NOT LOCAL: TYPE II SUBSEMIGROUPS CAN FALL ARBITRARILY IN COMPLEXITY

2006 ◽  
Vol 16 (04) ◽  
pp. 739-748 ◽  
Author(s):  
JOHN RHODES ◽  
BENJAMIN STEINBERG
Keyword(s):  
Finite Semigroup ◽  
Type Ii ◽  
Local Type

We prove the following two results announced by Rhodes: the Type II subsemigroup of a finite semigroup can fall arbitrarily in complexity; the complexity pseudovarieties Cn (n ≥ 1) are not local.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

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