On parastrophic algebras

Shaw, James Byrnie

doi:10.1090/s0002-9947-1915-1501018-x

Cited by 5 publications

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“…Typically, "conjugate" [149,150] is used in place of "parastrophe" [128,139]. Here we use "parastrophe" to match the author's PhD thesis [153].…”

Section: Isotopism and Parastrophymentioning

confidence: 99%

“…The term "parastrophe" is due to J. B. Shaw [139], but the concept goes back to E. Schröder [136] and even L. Euler [43]. Parastrophes have been considered by E. Schröder [136,137] and Schönhardt [134] under the name "Koordiniert", by R. H. Bruck [13,14] as the "associated" algebras, by I. M. H. Etherington [42] as "transposes", by H. A. Thurston [159] as "equasigroups" and by D. A. Norton and S. K. Stein [112,113,149,150] as "conjugates".…”

Section: Isotopism and Parastrophymentioning

confidence: 99%

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The Many Formulae for the Number of Latin Rectangles

Stones¹

2010

Electron. J. Combin.

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A $k \times n$ Latin rectangle $L$ is a $k \times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols. If $k=n$ then $L$ is a Latin square. Let $L_{k,n}$ be the number of $k \times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.

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“…Typically, "conjugate" [149,150] is used in place of "parastrophe" [128,139]. Here we use "parastrophe" to match the author's PhD thesis [153].…”

Section: Isotopism and Parastrophymentioning

confidence: 99%

Section: Isotopism and Parastrophymentioning

confidence: 99%

The Many Formulae for the Number of Latin Rectangles

Stones¹

2010

Electron. J. Combin.

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“…Concerning algebras, conjugacy was already dealt with by James Shaw [71] in 1915: Let A be an algebra over a base field F of basis {e 1 , . .…”

Section: Theoremmentioning

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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Quasigroupes parastrophiques. Expressions et identités

Sade¹

1959

Mathematische Nachrichten

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Sade, Quasigroupes parastrophiquesDefinition 1.1. Le conjoint du quasigroupe Q ( x ) est le quasigroupe C( -3c )[9, p. 601 dont la loi de composition est ddfinie par x x y = z + y * x = z .Si Q est fini, la table de CAYLEY de C derive de celle de Q par rabattement autour de la diagonale principale. Un quasigroupe self-conjoint est un quasigroupe commutatif. Definition 1.2. Le rdciproque de Q ( x ) est le quasigroupe R (0) [ 14, No 103, p. 131 ddfini par la division k droite x x y = 2 + 2 0 y = x. Definition 1.3. On peut dire que l'opdration (0) ddfinit le quotient gauche sur Q. U n quasigroupe est self-rdciproque s'il coancide avec son rdciproque. Si, it tout element (x, y ) de l'ensemble produit Q 2 , on fait correspondre le ((point)) dont les coordonnkes cartksiennes sont x, y , z, avec z = x x y , on obtient une representation gkometrique de Q au moyen d'un ensemble de points. Le rkciproque de Q correspond a un ensemble symetrique du premier par rapport au plan bissecteur du dikdre x o y z . Definition 1.4. L a division u gauche x x y = z~z @ x = y , fournit le quotient a droite sur Q. Les quasigroupes dont les opehtions sont ( * ), (@), et (0) sont dits parastrophiques de & ( x ) . Dkfintion 1.5. Les deux dernibres parastrophies correspondent nux opdrations (0) et (@), de'jinies par x x y = z + x o x = y , x X y = z + y @ Z = x .A chaque parastrophie correspond une permutation des coordonnkes x , y , z ; l'ensemble des six parastrophies est donc isomorphe au groupe symhtrique G'zyz . Comme celui-ci est engendre par deux de ses transpositions, on a la remarque suivante. Remarque 1.6. Toute propridtd d'un quasigroupe qui appartient simultandment au conjoint et a u rdciproque sera vraie de tous ses parastrophiques. Definition 1.7. Une expression sur Q est une suite d'dldments de Q , sdpards par les signea des six parastrophies ( X , 3c , 0, 0, 0, 0) et par un syst4me ddtermind de parenthhea, crochets et accolades . . . etc. Elle dkfinit une suite de calculs i L effectuer dans un ordre bien determine et dont le resultat final est un klement du quasigroupe. Deux expressions sont dgales si elles definissent le meme element.

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On parastrophic algebras

Cited by 5 publications

References 0 publications

The Many Formulae for the Number of Latin Rectangles

The Many Formulae for the Number of Latin Rectangles

A Historical Perspective of the Theory of Isotopisms

Quasigroupes parastrophiques. Expressions et identités

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