Autoparatopisms of Quasigroups and Latin Squares

Mendis, Mahamendige Jayama Lalani; Wanless, Ian M.

doi:10.1002/jcd.21515

Cited by 12 publications

(8 citation statements)

References 14 publications

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“…Computational methods for determining autotopism groups of partial Latin rectangles were compared in [17,18,69,29]. For Latin squares of order n 17, identifying when #PLR((Θ, π)) = 0 (throughout this paper # denotes the cardinality of a set) was done for isotopisms in [64] and paratopisms in [56], with prior work in [26,32].…”

Section: Introductionmentioning

confidence: 99%

Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

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Section: Introductionmentioning

confidence: 99%

Enumerating Partial Latin Rectangles

Ganfornina

Stones

2020

Electron. J. Combin.

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“…The same autotopism combines with (2, 1, 3)-conjugation to produce an autoparatopism, since L is symmetric. We used the classifications of autotopisms [20] and autoparatopisms [16]. From those lists we deduce that α and γ must have one of the cycle structures given in Table 3 and that we are at liberty to fix any choice of α and γ with the appropriate cycle structure.…”

Section: Symmetric Latin Squaresmentioning

confidence: 99%

Enumeration of Latin squares with conjugate symmetry

McKay

Wanless

2021

J of Combinatorial Designs

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A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order n equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order n + 1, and (2) suppose A and B are totally symmetric Latin squares of order ≢ n 0 mod 3.If A and B are paratopic then A and B are isomorphic.

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“…Recent advances about the sets of autotopisms, automorphisms and autoparatopisms of Latin squares are exposed, for instance, in [145][146][147][148]. We note in particular on the implementation of autotopisms of Latin squares into the design of authentication schemes [149], secret sharing schemes [150,151] and cryptographic transformations [152] in Cryptography.…”

Section: Quasigroups Latin Squares and Related Structuresmentioning

confidence: 99%

A Historical Perspective of the Theory of Isotopisms

2018

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In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

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Autoparatopisms of Quasigroups and Latin Squares

Cited by 12 publications

References 14 publications

Enumerating Partial Latin Rectangles

Enumerating Partial Latin Rectangles

Enumeration of Latin squares with conjugate symmetry

A Historical Perspective of the Theory of Isotopisms

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