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### Reducibility of pointlike problems

J. Almeida, J. C. Costa, M. Zeitoun
2015 Semigroup Forum
We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of  ...  Allowing only trivial groups, we obtain omega-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (A) and of all finite  ...  Hence V is σ-reducible for idempotent pointlike sets. Corollary 4.4.  ...

### Pointlike sets with respect to R and J

Jorge Almeida, José Carlos Costa, Marc Zeitoun
2008 Journal of Pure and Applied Algebra
We also give an algorithm to compute J-pointlike sets, where J denotes the pseudovariety of all finite J-trivial semigroups.  ...  We finally show that, in contrast with the situation for R, the natural adaptation of Henckell's algorithm to J computes pointlike sets, but not all of them.  ...  Henckell presented algorithms for computing A-pointlike sets [17] and A-idempotent pointlike sets [18] for the pseudovariety A of aperiodic semigroups.  ...

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

JOHN RHODES, BENJAMIN STEINBERG
2006 International journal of algebra and computation
Since H is V-idempotent pointlike, there is an idempotent e of T with H ≤ e ϕ −1 2 . Let q ∈ xϕ. Then xH ⊆ q ϕ −1 1 H ⊆ (q e )ϕ −1 1 . Setting q = q e completes the proof.  ...  Since H is a group and is V-pointlike, our above observations show that H is in fact V-idempotent pointlike. Set x = (a 0 , 1, 0 ), y = (a 0 , 1, 2 ). Notice that xH = yH = {x, y}.  ...

### Page 8909 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews
In the 1970s, Henckell and Rhodes introduced the notion of a pointlike set: For a pseudovariety V of monoids, a subset Y of a finite monoid S is said to be V-pointlike if for every relational morphism  ...  In order to prove his result, the author first generalizes the notion of pointlike sets to categories, studying how pointlikes for V and gV (the pseudovariety of categories generated by monoids in V viewed  ...

### A Profinite Approach to Stable Pairs [article]

Karsten Henckell, John Rhodes, Benjamin Steinberg
2006 arXiv   pre-print
We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids.  ...  Then the maximal V-idempotent pointlikes of M are precisely the maximal idempotents of PL V (M ).Proof. We already observed that idempotents of PL V (M ) are V-idempotent pointlike.  ...  Also the set of V-idempotent pointlikes of M form a downwards closed subset of P (M ). Next we consider the notion of a V-stable pair.  ...

### Page 3002 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews
The author considers the operation S ++ Ag(S), where Ag(S) is the set of all G-pointlike subsets of S, a semigroup under sub- set multiplication; and the associated operation on pseudovarieties of finite  ...  A subset ¥ of a finite semigroup S is group-pointlike, or G-pointlike, if for each relational morphism t: S — G into a finite group G, X¥ C r~'(g) for some g € G.  ...

### An application of a Theorem of Ash to finite covers

Karl Auinger, Gracinda M. S. Gomes, Victoria Gould, Benjamin Steinberg
2004 Studia Logica: An International Journal for Symbolic Logic
We show how the existence of nite proper covers for semigroups in this quasivariety is a consequence of Ash's celebrated theorem for pointlike sets. Date: November 7, 2001.  ...  satis ed for all elements a and idempotents e.  ...  We begin with the well known characterisation of pointlike sets (see Theorem 1.2 of 8]) which is a consequence of Ash's theorem (Theorem 2.1 in 1]) and the fact that in a semigroup in which the idempotents  ...

### Pointlike sets with respect to ER [article]

Karsten Henckell, Samuel Herman
2022 arXiv   pre-print
We show that pointlike sets are decidable for the pseudovariety of finite semigroups whose idempotent-generated subsemigroup is R-trivial.  ...  Notably, our proof is constructive: we provide an explicit relational morphism which computes the ER-pointlike subsets of a given finite semigroup.  ...  General theory of pointlike sets 4.1. In this section we will briefly cover key aspects of the authors' "general theory of pointlike sets" which provides the framework for our work here.  ...

### Some reducibility properties for pseudovarieties of the form DRH [article]

Célia Borlido
2015 arXiv   pre-print
The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability questions concerning pseudovarieties of the forms V * W, V join W, and  ...  idempotent pointlike equations (x 1 = · · · = x n = x 2 n ).  ...  The subsequent sections focus on pointlike, graph, and idempotent pointlike equations, in this order.  ...

### Some reducibility properties for pseudovarieties of the form DRH

Célia Borlido
2017 Communications in Algebra
The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability questions concerning pseudovarieties of the forms V * W, V ∨ W, and V  ...  Idempotent pointlike equations We now take for C the class of all systems of idempotent pointlike equations.  ...  idempotent pointlike equations (x 1 = · · · = x n = x 2 n ).  ...

### Page 4352 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews
In particular, it is proved that the join of a pseudovariety, for which pointlike sets are com- putable, with a pseudovariety, whose free objects are finite and computable, has again computable pointlike  ...  sets.  ...

### Pointlike reducibility of pseudovarieties of the form V ∗D

José Carlos Costa, Conceição Nogueira, Maria Lurdes Teixeira
2016 International journal of algebra and computation
We establish a connection between pointlike reducibility of V* D and the pointlike reducibility of the pseudovariety V.  ...  In particular, for the canonical signature κ consisting of the multiplication and the (ω-1)-power, we show that V* D is pointlike κ-reducible when V is pointlike κ-reducible.  ...  One says that V has decidable pointlikes if one can effectively compute all the V-pointlike sets of any given finite semigroup.  ...

### A General Theory of Pointlike Sets [article]

Karsten Henckell, Samuel Herman
2021 arXiv   pre-print
We introduce a general unifying framework for the investigation of pointlike sets.  ...  Notably, this provides a characterization of pointlikes which does not mention relational morphisms. Along the way, we formalize various common heuristics and themes in the study of pointlike sets.  ...  pointlike context EP W = 1 − Like W , which assigns S ∈ FinSgp to its set of W-idempotent pointlike subsemigroups, which are subsemigroups T of S such that whenever ρ : S W is a relational morphism with  ...

### Profinite Semigroups, Mal'cev Products, and Identities

J.-E Pin, P Weil
1996 Journal of Algebra
We also characterize the pointlike subsets of a finite semigroup by means of a relational morphism into a profinite semigroup.  ...  Abstract: We compute a set of identities defining the Mal'cev product of pseudovarieties of finite semigroups or finite ordered semigroups.  ...  Moreover, we verify that a subset X of a finite semigroup S is a V-pointlike if and only if it is a V -pointlike. Suppose indeed that X is a V -pointlike subset.  ...