## FREE ADEQUATE SEMIGROUPS (902)

Citations: | 6 - 1 self |

### BibTeX

@MISC{Kambites902freeadequate,

author = {Mark Kambites},

title = {FREE ADEQUATE SEMIGROUPS},

year = {902}

}

### OpenURL

### Abstract

Abstract. We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial “folding ” operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are J-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2, 1,1)-algebras have decidable word problem. 1.

### Citations

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Distance in graphs
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- 1990
(Show Context)
Citation Context ... retraction to a subtree is again a retraction. The following proposition is an instance of a well-known phenomenon, another important example of which is the uniqueness of the core of a finite graph =-=[13, 14]-=-. It can be deduced from very general results about morphisms of finite relational structures (for example [5, Proposition 1.4.7]) but for completeness we sketch a simple combinatorial proof. Proposit... |

100 |
Topology of finite graphs
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(Show Context)
Citation Context ...hich must happen, since each identification reduces the number of edges). This is essentially one part (since there is no deletion of spurs) of the “folding” process originally developed by Stallings =-=[17]-=- and extensively employed and extended to solve numerous problems in combinatorial group theory. Of course there is also a natural retract of the free adequate monoid onto the free monoid on the same ... |

67 |
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(Show Context)
Citation Context ...een of immense value in understanding inverse semigroups. Subsequently there have appeared a number of alternative proofs of Munn’s result and different representations for the free inverse semigroup =-=[10, 13, 14, 15]-=-. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups [6, 7, 8] and also in the closely related setting of Cockett-Lack rest... |

50 | The core of a graph
- Hell, Neˇsetˇril
- 1992
(Show Context)
Citation Context ... retraction to a subtree is again a retraction. The following proposition is an instance of a well-known phenomenon, another important example of which is the uniqueness of the core of a finite graph =-=[13, 14]-=-. It can be deduced from very general results about morphisms of finite relational structures (for example [5, Proposition 1.4.7]) but for completeness we sketch a simple combinatorial proof. Proposit... |

30 |
Adequate semigroups
- Fountain
- 1979
(Show Context)
Citation Context ...concerned with semigroups which admit local inverses with respect to nonidentity idempotents; chief among these are the regular and inverse semigroups. In the late 1970’s, it was observed by Fountain =-=[4, 5]-=- that many of the desirable properties of regular and inverse semigroups stem not directly from the existence of local inverses, but rather from the consequent fact that the cancellation properties of... |

30 |
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(Show Context)
Citation Context ...ss as a collection of equivalence classes of elements in a free algebra. In the case of inverse semigroups, a description of the free objects first discovered by Scheiblich [16] was developed by Munn =-=[12]-=- into an elegant geometric representation which has been of immense value in understanding inverse semigroups. Subsequently there have appeared a number of alternative proofs of Munn’s result and diff... |

29 | Restriction categories i: Categories of partial maps
- Cockett, Lack
(Show Context)
Citation Context ...previous remark, the free left adequate category will also be the free left Ehresmann category. Left Ehresmanm categories are generalisations of the restriction categories studied by Cockett and Lack =-=[2]-=-, which in the terminology of semigroup theory are weakly left E-ample categories [9]. The generalisation of our results to categories thus relates to our main results in the same way that the descrip... |

15 |
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(Show Context)
Citation Context ...e free adequate semigroups will also be free objects in any category of (2,1,1)-algebras which contain them and satisfy these conditions. This includes in particular the class of Ehresmann semigroups =-=[11]-=-. Similar remarks apply to free left adequate semigroups and free right adequate semigroups (which in particular are free objects also in the classes left Ehresmann semigroups and right Ehresmann semi... |

12 | Free ample monoids
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(Show Context)
Citation Context ...ferent representations for the free inverse semigroup [10, 14, 15, 16]. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups =-=[6, 7, 8]-=- and also in the closely related setting of Cockett-Lack restriction categories [1]. All of these techniques rely on certain identities satisfied by the classes of semigroups (or categories) in questi... |

11 |
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(Show Context)
Citation Context ...ress any members of the class as a collection of equivalence classes of elements in a free algebra. In the case of inverse semigroups, a description of the free objects first discovered by Scheiblich =-=[16]-=- was developed by Munn [12] into an elegant geometric representation which has been of immense value in understanding inverse semigroups. Subsequently there have appeared a number of alternative proof... |

10 | Universal algebra, volume 6 of Mathematics and its Applications - Cohn - 1981 |

9 |
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(Show Context)
Citation Context ...ferent representations for the free inverse semigroup [10, 13, 14, 15]. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups =-=[6, 7, 8]-=- and also in the closely related setting of Cockett-Lack restriction categories [1]. All of these techniques rely on certain identities satisfied by the classes of semigroups (or categories) in questi... |

3 |
A new construction for free inverse semigroups
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(Show Context)
Citation Context ...een of immense value in understanding inverse semigroups. Subsequently there have appeared a number of alternative proofs of Munn’s result and different representations for the free inverse semigroup =-=[10, 13, 14, 15]-=-. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups [6, 7, 8] and also in the closely related setting of Cockett-Lack rest... |

2 |
Cockett and Xiuzhan Guo. Stable meet semilattice fibrations and free restriction categories
- B
(Show Context)
Citation Context ...s approach have since been used to describe the free objects in a number of more general classes of semigroups [6, 7, 8] and also in the closely related setting of Cockett-Lack restriction categories =-=[1]-=-. All of these techniques rely on certain identities satisfied by the classes of semigroups (or categories) in question, which permit the rewriting of expressions to move idempotents to one side, and ... |

2 |
Free right h-adequate semigroups
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(Show Context)
Citation Context ...ferent representations for the free inverse semigroup [10, 13, 14, 15]. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups =-=[6, 7, 8]-=- and also in the closely related setting of Cockett-Lack restriction categories [1]. All of these techniques rely on certain identities satisfied by the classes of semigroups (or categories) in questi... |

2 |
Structure of left adequate and related monoids. Preprint available online at www-users.york.ac.uk/~varg1/adequate.pdf
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(Show Context)
Citation Context ...roups which satisfy the conditions defining adequate semigroups on one side only). An alternative approach to free left and right adequate semigroups appears in recent work of Branco, Gomes and Gould =-=[1]-=-.FREE ADEQUATE SEMIGROUPS 3 In addition to this introduction, this article comprises six sections. In Section 2 we briefly recall the definition of adequate semigroups, and summarise some of their el... |

2 | Homomorphisms and Structural Properties of Relational Systems - Foniok - 2007 |

1 |
Free ample monoids. Preprint available online at www-users.york.ac.uk/~varg1/freeample.ps
- Fountain, Gomes, et al.
- 2007
(Show Context)
Citation Context ...ferent representations for the free inverse semigroup [10, 13, 14, 15]. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups =-=[6, 7, 8]-=- and also in the closely related setting of Cockett-Lack restriction categories [1]. All of these techniques rely on certain identities satisfied by the classes of semigroups (or categories) in questi... |

1 | On free inverse semigroups
- Gutiérrez
(Show Context)
Citation Context ...een of immense value in understanding inverse semigroups. Subsequently there have appeared a number of alternative proofs of Munn’s result and different representations for the free inverse semigroup =-=[10, 13, 14, 15]-=-. Variants of Munn’s approach have since been used to describe the free objects in a number of more general classes of semigroups [6, 7, 8] and also in the closely related setting of Cockett-Lack rest... |

1 |
A new construction for free inverse semigroups
- Reynolds
- 1984
(Show Context)
Citation Context |

1 | Free left and right adequate semigroups
- Kambites
(Show Context)
Citation Context ...e [respectively, right adequate] semigroups. The proof, which is similar in spirit and outline to Section 5 but rather different in some of the technical details, will appear in a subsequent article. =-=[11]-=- The construction in Section 5 of a morphism from T 1 (Σ) to an adequate monoid M depends only on the facts that M is associative with commuting idempotents, and that the + and ∗ operations are idempo... |