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Register a new userParastrophes of quasigroups
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Authors
Wieslaw A. Dudek
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Parastrophes (conjugates) of a quasigroup can be divided into separate classes containing isotopic parastrophes. We prove that the number of such classes is always 1, 2, 3 or 6. Next we characterize quasigroups having a fixed number of such classes.
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Any pentagonal quasigroup is proved to have the product xy = R(x)+y-R(y) where (Q,+) is an Abelian group, R is its regular automorphism satisfying R^4-R^3+R^2-R+1 = 0 and 1 is the identity mapping. All abelian groups of order n<100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity (xy*x)y*x = y is proved to be the variety of commutative pentagonal quasigroups, whose spectrum is {11^n : n = 0,1,2,...}. We prove that the only translatable commutative pentagonal quasigroup is xy = (6x+6x)(mod11). The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the additive group Zn of integers modulo n and its automorphism R(x) = ax is proved to determine the value of a and the possible values of n.
The concept of a k-translatable groupoid is introduced. Those k-translatable quadratical quasigroups induced by the additive group of integers modulo m, where k<40, are listed for m<1200. The fine structure of quadratical quasigroups is explored in detail and the Cayley tables of quadratical quasigroups of orders 5, 9, 13 and 17 are produced. All but those of order 9 are k-translatable, for some k. Open questions and thoughts about future research in this area are given.
We prove the main result that a groupoid of order n is an idempotent k-translatable quasigroup if and only if its multiplication is given by x.y = (ax+by)(mod n), where a+b = 1(mod n), a+bk = 0(mod n) and (k,n)= 1. We describe the structure of various types of idempotent, k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent, translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical.
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We prove that quadratical quasigroups form a variety Q of right and left simple groupoids. New examples of quadratical quasigroups of orders 25 and 29 are given. The fine structure of quadratical quasigroups and inter-relationships between their properties are explored. The spectrum of Q is proved to be contained in the set of integers equal to 1 plus a multiple of 4.
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