Retracts of trees and free left adequate semigroups

Kambites, Mark

doi:10.1017/s0013091509001230

Cited by 13 publications

(13 citation statements)

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“…Our representation also gives rise to a nondeterministic polynomial time decision algorithm for the word problem in these semigroups; the exact computational complexity of the word problem remains for now unclear, and deserves further study. [3] Free adequate semigroups 367 We show elsewhere [16] that our approach also leads to a description of the free objects in the categories of left and right adequate semigroups (roughly speaking, those semigroups 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 et al [1,11].…”

Section: Introductionmentioning

confidence: 97%

Free Adequate Semigroups

Kambites

2011

J. Aust. Math. Soc.

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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.2010 Mathematics subject classification: primary 20M10; secondary 08A10, 08A50.

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Section: Introductionmentioning

confidence: 97%

Free Adequate Semigroups

Kambites

2011

J. Aust. Math. Soc.

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“…(Indeed it is an Ehresmann semigroup since there is an analogous range operation as well.) Free left Ehresmann semigroups are described in terms of trees in [13], and (in the monoid case) in terms of a generalised semidirect product construction involving monoids acting as orderpreserving maps on semilattices with identity in the sequence of papers [3] and [6].…”

Section: Definition 21 a D-semigroup Is Left Ehresmann If It Is D-sementioning

confidence: 99%

D-semigroups and constellations

Stokes

2017

Semigroup Forum

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In a result generalising the Ehresmann-Schein-Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories.Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations. (A general constellation is a one-sided generalisation of a category.) We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results.Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson's Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms.

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“…We regard Ehresmann semigroups as algebras with signature (2, 1, 1); as such, they form a variety E. Indeed, E is the variety generated by A, where A is the quasivariety of adequate semigroups [29]. The corresponding result is the one-side case may be found in [16] or [30]. An important property of Ehresmann semigroups is given below.…”

Section: Ehresmann Semigroupsmentioning

confidence: 99%

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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Retracts of trees and free left adequate semigroups

Cited by 13 publications

References 9 publications

Free Adequate Semigroups

Free Adequate Semigroups

D-semigroups and constellations

Beyond regular semigroups

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