Semigroup theory of symmetry

Rosenfeld, Vladimir R.; Nordahl, Thomas E.

doi:10.1007/s10910-016-0653-4

Cited by 7 publications

(5 citation statements)

References 92 publications

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“…We will next see that the maximum ordinarization number stated in the previous lemma is attained uniquely by the numerical semigroup in (1). To prove this result, we will need the next lemma.…”

Section: Lemma 4 the Maximum Quasi-ordinarization Number Of A Non-ordinary Semigroup Of Genus G Is G−1mentioning

confidence: 82%

“…Define, as in [7], the sequence f ω by f ω = ∑ Ω∈S ω #C(Ω, ω + 1). The first elements in the sequen from f 0 to f 15 are We remark that this sequence appears in [1], where Bernardini and Torres proved that the numb of numerical semigroups of genus 3ω whose number of even gaps is ω is exactly f ω . It corresponds the entry A210581 of The On-Line Encyclopedia of Integer Sequences [23].…”

Section: Lemmamentioning

confidence: 93%

“…A number of papers deal with the sequence n g [1][2][3]6,7,[9][10][11]13,[16][17][18][19][20]22,28]. Alex Zhai proved the asymptotic Fibonacci-like behavior of n g [27].…”

Section: Figurementioning

confidence: 99%

“…While the symmetry of structures has traditionally been studied with the aid of groups, it is also possible to relax the definition of symmetry, so as to describe some forms of symmetry that arise in quasicrystals, fractals, and other natural phenomena, with the aid of semigroups or monoids, rather than groups. For example, Rosenfeld and Nordahl [1] lay the groundwork for such a theory of symmetry based on semigroups and monoids, and they cite some applications in chemistry.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-Ordinarization Transform of a Numerical Semigroup

2021

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In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical semigroups of each given genus. We analyze the number of nodes at each depth in the forest and propose new conjectures. Some properties of the quasi-ordinarization transform are presented, as well as some relations between the ordinarization and quasi-ordinarization transforms.

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Section: Lemma 4 the Maximum Quasi-ordinarization Number Of A Non-ordinary Semigroup Of Genus G Is G−1mentioning

confidence: 82%

Section: Lemmamentioning

confidence: 93%

“…A number of papers deal with the sequence n g [1][2][3]6,7,[9][10][11]13,[16][17][18][19][20]22,28]. Alex Zhai proved the asymptotic Fibonacci-like behavior of n g [27].…”

Section: Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasi-Ordinarization Transform of a Numerical Semigroup

2021

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“…The case D = K 4 is a special case of an unweighted divisor, which is a simple graph; but in general, divisors are arbitrarily weighted graphs or can also be multigraphs if their adjacency matrix S is a nonnegative integer matrix. Add that a particular case of the equitable partition of graph vertices is their partition into orbits induced by the automorphism (symmetry) group [33] or endomorphism monoid [34] of the graph.…”

Section: Preliminariesmentioning

confidence: 99%

Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph

Rosenfeld

2024

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The vertex quadrangulation QG of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph QG include the full spectrum {3,(−1)3} of the tetrahedron graph (complete graph K4)? Previously, many bipartite and nonbipartite graphs QG with such a subspectrum have been found; for example, a nonbipartite variant of the graph QK5. Here, we present one of the variants of the nonbipartite vertex quadrangulation QO of the octahedron graph O, which has eigenvalue (−1) of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant QO contains eigenvalue (−1) of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph QG. Here, we continue to explore the spectrum of graphs QG. Some possible connections of the mathematical theme to chemistry are also noted.

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Urban–Urban Disparities

Cheung

2022

Statecraft in Symbols

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Semigroup theory of symmetry

Cited by 7 publications

References 92 publications

Quasi-Ordinarization Transform of a Numerical Semigroup

Quasi-Ordinarization Transform of a Numerical Semigroup

Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph

Urban–Urban Disparities

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