On three approaches to conjugacy in semigroups

Kudryavtseva, Ganna; Mazorchuk, Volodymyr

doi:10.1007/s00233-008-9047-7

Cited by 31 publications

(37 citation statements)

References 13 publications

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“…There has been a number of attempts to dene conjugate elements in semigroups, generalising conjugation in groups. For recent reviews of ideas in this direction, see [4,5]. In this context, knot semigroups are interesting as examples of semigroups where conjugacy is introduced explicitly: indeed, each pair of relations dening a knot semigroup states precisely that two elements x, z in the semigroup are conjugate.…”

Section: The Context and The Paper Planmentioning

confidence: 99%

“…In general, we do not know yet how to produce a presentation re-dening a given knot semigroup which has been dened by a cancellative presentation 3 . Employing cancellation is natural because it ensures that the knot semigroup is preserved by the rst Reidemeister move 4 . However, the knot semigroup is not preserved by neither the second nor the third Reidemeister move; see Section 8 for more details and a discussion.…”

Section: Main Definitionsmentioning

confidence: 99%

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Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and $$yx=zy$$ y x = z y and connections with knot theory

Vernitski

2016

Semigroup Forum

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Abstract. We introduce a knot semigroup as a cancellative semigroup whose dening relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we dene it is closely related to such tools of knot theory as the 2-fold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T (2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally dened factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research. The context and the paper planWe consider cancellative semigroups (which we call knot semigroups) whose dening relations come in pairs of the form xy = yz and yx = zy, where x, y, z are generators, and are`read' from a certain natural diagram (namely, a knot diagram 1 ). Our inspiration in this research comes partially from the study of right-angled Artin groups (and the corresponding semigroup-theory construction, trace monoids [1]). A right-angled Artin group is a group in which every dening relation has a form xy = yx. Given an undirected graph, one can dene a group whose set of generators is the set of vertices of the graph, and a dening relation xy = yx is introduced whenever vertices x and y are adjacent. This construction denes a natural correspondence between undirected graphs and right-angled Artin groups. Whereas knot diagrams are not as ubiquitous as graphs, they have attracted much attention of algebraists in the last century, and knot semigroups described in this paper can become a new natural way of dening semigroups corresponding to knot diagrams; we discuss this further in Section 8. Each relation dening a knot semigroup has words of the same length on the two sides of the equality; such relations are called homogeneous and, accordingly, semigroups dened in this way are also sometimes called homogeneous; another example of homogeneous semigroups are braid semigroups (for their denition see, for example, [2]); for a brief review of more examples of classes of homogeneous semigroups see [3]. There has been a number of attempts to dene conjugate elements in semigroups, generalising conjugation in groups. For recent reviews of ideas in this direction, see 1 Note that a knot semigroup is not a knot invariant; that is, there are cases when two dierent diagrams of the same knot produce non-isomorphic knot semigroups. See more in Section 8.

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Section: The Context and The Paper Planmentioning

confidence: 99%

Section: Main Definitionsmentioning

confidence: 99%

Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and $$yx=zy$$ y x = z y and connections with knot theory

Vernitski

2016

Semigroup Forum

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“…The following definition for conjugacy in a semigroup S is taken from [9]. In [14] it is referred to as the transitive closure of transposition, however in this article we will refer to it simply as conjugacy.…”

Section: Conjugacy and Diagramsmentioning

confidence: 99%

A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

Cummings

Jackson

2017

Semigroup Forum

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There are multiple, inequivalent, definitions for conjugacy in semigroups. In Cummings and Jackson (Semigroup Forum 88, 52-66, 2014), we conjectured that, for at least one of these definitions of conjugacy, the conjugacy problem for finitely presented semigroups satisfying C(2) and T (4) is solvable. Here we essentially verify that conjecture. In that 2014 Semigroup Forum publication, we developed geometric methods to solve a conjugacy problem for finitely presented semigroups satisfying C(3). We use those methods again here.

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“…Since multiplication is associative, this gives rise to a possibly infinite semigroup H generated by diagrams and their products. Following [13], we define and completely describe three notions of conjugacy in H extending results from [12].…”

Section: Introductionmentioning

confidence: 99%

Conjugation in Brauer algebras and applications to character theory

Shalile¹

2011

Journal of Pure and Applied Algebra

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a b s t r a c tThe Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.In particular, there is an ordering of simple and cell modules such that D B r (δ) is lower triangular with diagonal entries equal to 1. Remark 1.10. When F is a field of characteristic 0, then the decomposition matrix of the Brauer algebra was determined by Martin, see [16]. If the characteristic of F is 0 or larger than p, then an algorithm to compute the decomposition matrix of the Brauer algebra can also be found in [23]

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On three approaches to conjugacy in semigroups

Cited by 31 publications

References 13 publications

Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and $$yx=zy$$ y x = z y and connections with knot theory

Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and $$yx=zy$$ y x = z y and connections with knot theory

A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

Conjugation in Brauer algebras and applications to character theory

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