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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.
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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.
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.
The concept of a k-translatable groupoid is explored in depth. Some properties of idempotent k-translatable groupoids, left cancellative k-translatable groupoids and left unitary k-translatable groupoids are proved. Necessary and sufficient conditions are found for a left cancellative k-translatable groupoid to be a semigroup. Any such semigroup is proved to be left unitary and a union of disjoint copies of cyclic groups of the same order. Methods of constructing k-translatable semigroups that are not left cancellative are given.
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.
Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.
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