Maximal Regular Subsemibands of Singn

Yang, Xiuliang; Hao-bo, Yang

doi:10.1007/s00233-005-0103-2

Cited by 9 publications

(4 citation statements)

References 8 publications

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“…T e (T e (v)) = T e (v) for all e ∈ E. The elementary collapsing maps generate the transformation semigroup S(Γ) = T e |e ∈ E acting on Y, which is known as the flow semigroup or Rhodes semigroup of Γ [22]. It can be shown that T (y,y ′ ) ∈ S(Γ) if and only if (y, y ′ ) ∈ E or (y ′ , y) is contained in a directed cycle of Γ [23]. The flow semigroup therefore completely determines the undirected graph underlying Γ [22,24,25].…”

Section: The Flow Semigroupmentioning

confidence: 99%

Optimisation via encodings: a renormalisation group perspective

Klemm,

Mehta,

Stadler

2023

J. Phys. A: Math. Theor.

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Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.

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Section: The Flow Semigroupmentioning

confidence: 99%

Optimisation via encodings: a renormalisation group perspective

Klemm,

Mehta,

Stadler

2023

J. Phys. A: Math. Theor.

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“…Several authors have classified those digraphs D such that D has a specific semigroup property. For instance, in [YY06] those digraphs D such that D is regular are classified; and in [CCGM16] those D where D is a band are classified. In [YY06,YY09], necessary and sufficient conditions on digraphs D and D ′ are given so that D = D ′ or D ∼ = D ′ , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, in [YY06] those digraphs D such that D is regular are classified; and in [CCGM16] those D where D is a band are classified. In [YY06,YY09], necessary and sufficient conditions on digraphs D and D ′ are given so that D = D ′ or D ∼ = D ′ , respectively. In this paper, we continue in this direction, by classifying those digraphs D for which the semigroup D has one of a variety of properties.…”

Section: Introductionmentioning

confidence: 99%

Structural aspects of semigroups based on digraphs

East¹,

Gadouleau²,

Mitchell³

2019

Algebraic Combinatorics

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Given any digraph D without loops or multiple arcs, there is a natural construction of a semigroup D of transformations. To every arc (a, b) of D is associated the idempotent transformation (a → b) mapping a to b and fixing all vertices other than a. The semigroup D is generated by the idempotent transformations (a → b) for all arcs (a, b) of D.In this paper, we consider the question of when there is a transformation in D containing a large cycle, and, for fixed k ∈ N, we give a linear time algorithm to verify if D contains a transformation with a cycle of length k. We also classify those digraphs D such that D has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is K -trivial or K -universal where K is any of Green's H -, L -, R-, or J -relation, and when D has a left, right, or two-sided zero.

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“…The semigroup of non-decreasing transformations OI n := {α ∈ Sing n : v ≤ vα} is arc-generated by the transitive tournament T n on [n] (Figure 1 illustrates T 5 ). Connections between subsemigroups of Sing n and digraphs have been studied before (see [11,12,13,14]). The following definition, which we shall adopt in the following sections, appeared in [14]: We introduce some further notation.…”

Section: Introductionmentioning

confidence: 99%

Lengths of words in transformation semigroups generated by digraphs

et al. 2016

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Given a simple digraph D on n vertices (with n ≥ 2), there is a natural construction of a semigroup of transformations D . For any edge (a, b) of D, let a → b be the idempotent of rank n − 1 mapping a to b and fixing all vertices other than a; then, define D to be the semigroup generated bybe the minimal length of a word in E(D) expressing α. It is well known that the semigroup Sing n of all transformations of rank at most n − 1 is generated by its idempotents of rank n − 1. When D = K n is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate (K n , α), for any α ∈ K n = Sing n ; however, no analogous non-trivial results are known when D = K n . In this paper, we characterise all simple digraphs D such that either (D, α) is equal to Howie-Iwahori's formula for all α ∈ D , or (D, α) = n − fix(α) for all α ∈ D , or (D, α) = n − rk(α) for all α ∈ D . We also obtain bounds for (D, α) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n − 1 of Sing n ). We finish the paper with a list of conjectures and open problems.

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Maximal Regular Subsemibands of Singn

Cited by 9 publications

References 8 publications

Optimisation via encodings: a renormalisation group perspective

Optimisation via encodings: a renormalisation group perspective

Structural aspects of semigroups based on digraphs

Lengths of words in transformation semigroups generated by digraphs

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