Latin squares, partial latin squares and its generalized quotients (original) (raw)
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Small latin squares, quasigroups, and loops
Journal of Combinatorial Designs, 2007
We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, , quasigroups of order 6 (Bower, 2000) and loops of order 7 . The loops of order 8 have been independently found by " QSCGZ" and Guérin (unpublished, 2001).
2006
Abstract LetX be a n-set and letA= a ij be an xn matrix for whichaij⊆ X, for 1≤ i, j≤ n. A is called a generalized Latin square onX, if the following conditions is satisfied: ∪ _ i= 1^ n a_ ij= X= ∪ _ j= 1^ n a_ ij. In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a H v-structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of ordern, has a transversal set.
Algebraic Structures and Variations: From Latin Squares to Lie Quasigroups
2021
In this Master\u27s Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups
On certain constructions for latin squares with no latin subsquares of order two
Discrete Mathematics, 1976
A I4tm slluarc 1s said to be an N,-latm square (see [ 1) and 121) if it contains no latin subsquare of crrder 2. The exwence of KJatin squares of all orders except 2' has been proved in [2]. Twially. there are no such squares of orders 2 and 4. M. Mclxish [3) has shown that there exist S,-latin yuares of all orders 2' for k 3 6. The present paper introduces a constructictn for .V,-latin squares of all e\cn order\ n wth n f 0 (mad 1) and n f 3 (mod 5). The problem is thus solved fw the ,vdcr\ 2' and 1'. For 2'. the only remamirrg ~a~'. Eric Rcgencr of the Faculty of Music, L'niversit6 de Mont&L hers wnwuitcd the follcb*utg example of an X,-farm square and kindly granted us the permission to reproduce it here: The existence prcrhlrm of X_.-latln squaw\ IP thus completely solved.
Orthogonality and extendability of latin squares and related structures
2012
Two of the most important topics in the study of latin squares are questions of orthogonality and extendability. A latin square of order n is an n×n array consisting of n distinct symbols such that each of n symbols occurs precisely once in each row and column. Two latin squares are said to be orthogonal if no two cells contain List of Tables vi List of Symbols vii
Algebra and discrete mathematics, 2018
We introduce the notion of a Gorenstein Latin square and consider loops and quasigroups related to them. We study some properties of normalized Gorenstein Latin squares and describe all of them with order n ≤ 8.
The fine structures of three Latin squares
Journal of Combinatorial …, 2006
Denote by F in(v) the set of all integer pairs (t, s) for which there exist three Latin squares of order v on the same set having fine structure (t, s). We determine the set F in(v) for any integer v ≥ 9.