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Loops with exponent three in all isotopes. (English) Zbl 1337.20075

In this paper the results and proofs of the authors are a mix of the quasigroups and loops and Latin squares perspectives. From the quasigroup perspective, the authors are interested in properties that hold in all loop isotopes. The Latin square properties in which the authors are interested in are those invariant at the very least, under isotopy, that is, under permutations of the rows, of the columns and of the symbols.
G. H. J. van Rees has shown [in Ars Comb. 29B, 193-204 (1990; Zbl 0718.05014)] that a Latin square of order \(n\) has at most \(n^2(n-1)/18\) Latin subsquares of order 3. The authors show in this paper that it can only be achieved if \(n\equiv3\pmod 6\). The paper concentrates on the number of subsquares of order 3. The main result of this paper is Theorem 1.1, in Sec 1. In Sec 4, the authors examine a conjecture of van Rees regarding the possible orders of van Rees Latin squares, and show that such a square has order congruent to \(3\pmod 6\). In Sec 6, the authors show that on the underlying set of a van Rees loop there is a natural Steiner quasigroup structure (Theorem 6.3).

MSC:

20N05 Loops, quasigroups
05B15 Orthogonal arrays, Latin squares, Room squares

Citations:

Zbl 0718.05014

Software:

Prover9; GAP; Mace4; LOOPS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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