Free Adequate Semigroups

Kambites, Mark

doi:10.1017/s1446788711001753

Cited by 23 publications

(65 citation statements)

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“…We shall need a proposition, the essence of which is that the distinction between semigroups and monoids is unimportant. The proof is essentially the same as for the corresponding result in the (two-sided) adequate case, which can be found in [10].…”

Section: Corollary 25 Any Subset Of T 1 (σ) Closed Under the Operatmentioning

confidence: 61%

“…The subset Σ (which generates F ) is called a free generating set for F , and its cardinality is the rank of F . The following theorem was the main result of [10].…”

Section: Theorem 22 the Pruning Mapmentioning

confidence: 96%

“…The following proposition recalls some basic properties of left and right adequate semigroups; these are well known and full proofs can be found in [10]. …”

Section: Theorem 22 the Pruning Mapmentioning

confidence: 99%

“…In [10] we introduced and studied an intriguing multiplication operation on a class of edge-labelled, directed combinatorial trees. This multiplication will be described in detail below but, roughly speaking, two trees are combined by identifying a distinguished end vertex on the first tree with a distinguished start vertex on the second and then taking a minimal retract (or core, in the language of graph theory).…”

Section: Introductionmentioning

confidence: 99%

“…We showed in [10] that the set of all such trees (with labels from some fixed alphabet) is a naturally arising algebraic structure: it is a free object (of rank the cardinality of the labelling alphabet) in the class of adequate semigroups. Adequate semigroups, which were introduced by Fountain [5] in the 1970s, are semigroups in which the cancellation properties of elements in general are encapsulated in the cancellation properties of idempotent elements.…”

Section: Introductionmentioning

confidence: 99%

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Retracts of trees and free left adequate semigroups

Kambites

2011

Proceedings of the Edinburgh Mathematical Society

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Recent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

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Section: Corollary 25 Any Subset Of T 1 (σ) Closed Under the Operatmentioning

confidence: 61%

“…The subset Σ (which generates F ) is called a free generating set for F , and its cardinality is the rank of F . The following theorem was the main result of [10].…”

Section: Theorem 22 the Pruning Mapmentioning

confidence: 96%

“…The following proposition recalls some basic properties of left and right adequate semigroups; these are well known and full proofs can be found in [10]. …”

Section: Theorem 22 the Pruning Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Retracts of trees and free left adequate semigroups

Kambites

2011

Proceedings of the Edinburgh Mathematical Society

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Beyond regular semigroups

Wang

2015

Semigroup Forum

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The topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasiadequate, concordant, abundant, restriction, Ehresmann and weakly abundant semigroups. A semigroup S with subset of idempotents U is weakly U-abundant if every R U -class and every L U -class contains an idempotent of U, where R U and L U are relations extending the well known Green's relations R and L. We assume throughout that our semigroups satisfy a condition known as the Congruence Condition (C).We take several approaches to weakly U-abundant semigroups. Our first results describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly Uabundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectangular bands of monoids.The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly B-orthodox semigroups, where B is a band. We note that if B is a semilattice then a weakly B-orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly B-orthodox semigroup S as a spined product of a fundamental weakly Borthodox semigroup S B (depending only on B) and S/γ B , where B is isomorphic to B and γ B is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form δ B for γ B , which simplifies our result to a straightforward form.i For the above to work smoothly in the case S is weakly B-superabundant, we need to find a canonical fundamental weakly B-superabundant subsemigroup of S B . This we do, and give the corresponding answers in the case of the Hall semigroup W B and a number of intervening semigroups.We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a regular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad's work to weakly U-regular semigroups, that is, weakly U-abundant semigroups with (C) and U generating a regular subsemigroup whose set of idempotents is U.As an intervening step we consider weakly B-orthodox semigroups in this light. We take two approaches. The first is via a new construction of an inductive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids.We show that the category of weakly B-orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is influenced by that of Nambooripad, however, there are significant differences in strateg...

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