## Profinite semigroups, Mal'cev products and identities (1996)

Venue: | J. ALGEBRA |

Citations: | 39 - 17 self |

### BibTeX

@ARTICLE{Pin96profinitesemigroups,,

author = {J.-E. Pin and P. Weil},

title = {Profinite semigroups, Mal'cev products and identities},

journal = {J. ALGEBRA},

year = {1996},

volume = {182},

pages = {604--626}

}

### Years of Citing Articles

### OpenURL

### Abstract

We compute a set of identities defining the Mal'cev product of pseudovarieties of finite semigroups or finite ordered semigroups. We also characterize the pointlike subsets of a finite semigroup by means of a relational morphism into a profinite semigroup. Finally, we apply our results to the proof of the decidability of the Mal'cev products of a decidable pseudovariety with the pseudovarieties of nilpotent and of J -trivial semigroups.

### Citations

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Finite Semigroups and Universal Algebra, World Scienti
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Citation Context ...quences are explored in another paper by the authors [19]. The two main ingredients in our proofs are the following: the theory of relatively free profinite semigroups, as it was developed by Almeida =-=[2]-=- and by Almeida and the second author [4] in order to build upon Reiterman's theorem; and a lemma by Hunter on the properties of certain congruences in compact semigroups [13]. The paper is organized ... |

72 |
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Citation Context ... algorithm to decide membership in KH (S), then V fl m H is decidable. Deciding membership in KH (S) is a difficult question in general. Ash proved that KG (S) is decidable for any finite semigroup S =-=[7]-=-. More precisely, we say that a subset T of S is closed under weak conjugacy if sT t ` T and tTs ` T for all s; t 2 S such that sts = s. Ash proved that KG (S) is the smallest submonoid of S closed un... |

70 |
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Citation Context ...ts elements can be seen as limits of sequences of words on the alphabet A. An important such limit is the !-power, which traditionally denotes the idempotent power of an element of a finite semigroup =-=[8, 16]-=-. Proposition 1.3 Let A be a profinite set and let V be a pseudovariety of semigroups. Let x 2sFA (V). The sequence x n! converges insFA (V). Its limit, written x ! , is idempotent and, if oe: FA (V) ... |

69 | A variety theorem without complementation - Pin - 1995 |

67 |
The Birkhoff theorem for finite algebras, Algebra Universalis 14
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Citation Context ...ds. The Mal'cev product is a very important operation in the lattice of pseudovarieties, with applications in group theory, in semigroup theory and in language theory. It was established by Reiterman =-=[20]-=- that each pseudovariety is defined by a set of identities in some profinite structure. However, the identity problem, i.e. the problem of finding a defining set of identities for a pseudovariety, is ... |

51 |
On the profinite topology on a free group
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Citation Context ...gacy if sT t ` T and tTs ` T for all s; t 2 S such that sts = s. Ash proved that KG (S) is the smallest submonoid of S closed under weak conjugacy. Ribes and Zalesskii gave a new proof of this result =-=[21]-=-, and later proved the decidability of KGp (S) for any prime p (where G p is the pseudovariety of p-groups) [22]. Recently, Margolis, Sapir and Weil proved the decidability of KG nil (S), where G nil ... |

46 |
Relatively free profinite monoids: an introduction and examples
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- 1995
(Show Context)
Citation Context ...the authors [19]. The two main ingredients in our proofs are the following: the theory of relatively free profinite semigroups, as it was developed by Almeida [2] and by Almeida and the second author =-=[4]-=- in order to build upon Reiterman's theorem; and a lemma by Hunter on the properties of certain congruences in compact semigroups [13]. The paper is organized as follows. The first section contains th... |

44 | Ash’s type-II theorem, profinite topology and Malcev products
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Citation Context ...mma1 . The problem of the effective characterization of the V-pointlike subsets of a finite semigroup (resp. monoid) is closely related to the membership problem of the varieties of the form W fl m V =-=[9, 10, 11]-=-. Using quite the same method as for Theorem 3.1, we obtain the following result. Theorem 3.3 Let V be a pseudovariety of semigroups, let A be a profinite set and let S be a finite semigroup. Let oe: ... |

32 | Polynomial closure and unambiguous product
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Citation Context ...8.1.7]. The other computations are immediate applications of Theorem 5.10. ut Our last example deals with the pseudovariety B 1 , which plays an important role in the study of the dot-depth hierarchy =-=[19]-=-. By definition, B 1 = [[(x ! sy ! tx ! ) ! x ! sy ! vx ! (x ! uy ! vx ! ) ! = (x ! sy ! tx ! ) ! (x ! uy ! vx ! ) ! ]]: Proposition 5.13 B 1 fl m Nil = B 1 and B 1 fl m J 1 = B 1 fl m J. Proof. It is... |

30 | Profinite categories and semidirect products
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Citation Context ...o be very difficult, even for the simple operation of join of two pseudovarieties [1]. Recent work of Almeida and the second author addressed the problem for the semidirect product of pseudovarieties =-=[5, 6]-=-. In this paper, we solve the analogous problem for the Mal'cev product, that is, we describe a defining set of identities for a Mal'cev product. Our methods also apply to the case of pseudovarieties ... |

23 | Free profinite semigroups over semidirect products - Almeida, Weil - 1995 |

23 |
Pointlike sets: the finest aperiodic cover of a finite semigroup
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Citation Context ... [1]. The question of the decidability of a Mal'cev product is very difficult in general. In order to address this question, Henckell and Rhodes introduced the pointlike subsets of a finite semigroup =-=[9]-=-. Along with our result on the identities defining a Mal'cev product, we give a general theorem describing the pointlike subsets of a finite semigroup. Our results also lead to specific decidability r... |

21 | A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35
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- 1996
(Show Context)
Citation Context ...methods also apply to the case of pseudovarieties of finite ordered semigroups. Such pseudovarieties can also be defined by pseudoidentities in a profinite structure as was established by the authors =-=[18]-=-. Let us say immediately that our solution is not effective: even if the pseudovarieties V and W have decidable membership problems (we say that such pseudovarieties are decidable), the defining set o... |

21 |
The pro-p topology of a free group and algorithmic problems in semigroups
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Citation Context ...d of S closed under weak conjugacy. Ribes and Zalesskii gave a new proof of this result [21], and later proved the decidability of KGp (S) for any prime p (where G p is the pseudovariety of p-groups) =-=[22]-=-. Recently, Margolis, Sapir and Weil proved the decidability of KG nil (S), where G nil is the pseudovariety of nilpotent groups [14]. Let us also note the following proposition on membership in KH (S... |

20 |
Undecidability of the identity problem for finite semigroups
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- 1992
(Show Context)
Citation Context ...owever, the identity problem, i.e. the problem of finding a defining set of identities for a pseudovariety, is known to be very difficult, even for the simple operation of join of two pseudovarieties =-=[1]-=-. Recent work of Almeida and the second author addressed the problem for the semidirect product of pseudovarieties [5, 6]. In this paper, we solve the analogous problem for the Mal'cev product, that i... |

15 | Ash's type II theorem, pro topology and Malcev products - Henckell, Margolis, et al. - 1991 |

14 |
Certain finitely generated compact zero dimensional semigroups, J.Austral
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- 1988
(Show Context)
Citation Context ... was developed by Almeida [2] and by Almeida and the second author [4] in order to build upon Reiterman's theorem; and a lemma by Hunter on the properties of certain congruences in compact semigroups =-=[13]-=-. The paper is organized as follows. The first section contains the necessary definitions and results on free profinite semigroups and on identities, as well as Hunter's lemma. Section 2 introduces Ma... |

12 |
Some results on the dot-depth hierarchy, Semigroup Forum 46
- Weil
- 1993
(Show Context)
Citation Context ...J 1 ) = P(A) is called the content morphism. Note that if u 2 A is a word, then u is the alphabetic content of u, that is, u is the set of letters occurring in u. The following result was observed in =-=[24]-=- in the unordered case. Theorem 5.1 Let V be a pseudovariety of semigroups (resp. ordered semigroups), let A be a finite alphabet, let S be a finite semigroup (resp. ordered semigroup) and let oe: A +... |

10 | Semigrupos Finitos e Álgebra Universal, Publicaçoes do Instituto de Matematica e Estatística da Universidade de Saõ Paulo - Almeida - 1992 |

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9 |
Theorems on compact totally disconnected semigroups and lattices
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Citation Context ..., we say that a semigroup (resp. monoid) is pro-V if it is a projective limit of elements of V. It is known that a semigroup (resp. monoid) is profinite if and only if it is compact and 0-dimensional =-=[15]-=-. If V is a pseudovariety of semigroups (resp. monoid), then a profinite semigroup (resp. monoid) is pro-V if and only if all its finite continuous homomorphic images are in V. Of course, all elements... |

9 | Relatively free pro monoids: an introduction and examples - Almeida, Weil - 1995 |

7 | A reiterman theorem for pseudovarieties of structures. Algebra Universalis - Pin, Weil - 1996 |

6 |
On regular implicit operations, Portugaliae Mathematica 50
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Citation Context ... Nil. Let us now consider products of the form V fl m J, where J is the pseudovariety of finite J -trivial semigroups. It is well-known that J = Nil fl m J 1 [16]. Almeida [2] and Almeida and Azevedo =-=[3]-=- gave a detailed study of the structure of the free pro-J semigroups. Proposition 5.9 Let x; y 2 d A + . (1) J satisfies x ! = y ! if and only if x = y. (2) J satisfies x = x ! if and only if x = uv !... |

6 |
On the decidability of the membership problem of the pseudovariety
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- 1994
(Show Context)
Citation Context ... FA (V) the projective limit of the A-generated elements of V. The topological semigroups (resp. monoids) of the form FA (V) have been widely studied, see Almeida [2], Almeida and Weil [4] or Zeitoun =-=[26]-=-. The semigroup (resp. monoid)sFA (V) can be viewed as the completion of a certain uniform structure on the free semigroup A + (resp. free monoid A ), or as the semigroup (resp. monoid) of A-ary impli... |

6 | The algebra of implicit operations - Almeida - 1989 |

5 | Eilenberg’s theorem for positive varieties of languages - PIN - 1995 |

5 | Free pro semigroups over semidirect products, Izvestiya VUZ Matematika 39 - Weil - 1995 |

5 | Pointlike sets: the aperiodic cover of a semigroup - Henckell - 1988 |

3 |
Product expansions
- Henckell
- 1995
(Show Context)
Citation Context ...mma1 . The problem of the effective characterization of the V-pointlike subsets of a finite semigroup (resp. monoid) is closely related to the membership problem of the varieties of the form W fl m V =-=[9, 10, 11]-=-. Using quite the same method as for Theorem 3.1, we obtain the following result. Theorem 3.3 Let V be a pseudovariety of semigroups, let A be a profinite set and let S be a finite semigroup. Let oe: ... |

3 | Undecidability of the identity problem for semigroups - Albert, Baldinger, et al. - 1992 |

2 |
On the extension problem for finite inverse automata and closed subgroups in profinite topologies
- Margolis, Sapir, et al.
(Show Context)
Citation Context ...p (S) for any prime p (where G p is the pseudovariety of p-groups) [22]. Recently, Margolis, Sapir and Weil proved the decidability of KG nil (S), where G nil is the pseudovariety of nilpotent groups =-=[14]-=-. Let us also note the following proposition on membership in KH (S). Proposition 5.3 Let S be a finite monoid, let oe: c A ! S be a continuous morphism and let x 2 c A . If H satisfies x = 1, then xo... |

2 | The join of the pseudovarieties of idempotent semigroups and of locally trivial semigroups, Semigroup Forum
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- 1995
(Show Context)
Citation Context .... xsy) by Nil-substitution. The theorem now follows from Theorem 4.1. ut We derive from Theorem 5.5 the following results, the first of which is well known [2], and the third of which can be found in =-=[25]-=-. Corollary 5.6 The following equalities hold. (1) G fl m Nil = [[x ! = y ! ]]. (2) Com fl m Nil = [[xy ! z = zy ! x]]. (3) [[x = x 2 ]] fl m Nil = [[xy ! z = (xy ! z) 2 ]] = [[xy ! z = (xy ! z) ! ]].... |

1 | Certain generated compact zero-dimensional semigroups - Hunter - 1988 |

1 | On the extension problem for inverse automata and closed subgroups in pro topologies - Margolis, Sapir, et al. |