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Abstract
The concepts of transversal complete mapping are fundamental to the theory of Latin squares. This chapter summarizes the basic facts, generalizations, and results about these concepts. A transversal of a Latin square of order n is a set of n cells, one in each row, one in each column, such that no two of the cells contain the same symbol. If L is a Latin square, which satisfies the quadrangle criterion (that is, is isotopic to the multiplication table of a group) and, which possesses at least one transversal, then L has decomposition into n disjoint transversals and consequently, has an orthogonal mate. A loop is called a “Bol loop” if, for all elements x , y , z , [( xy ) z ] y = x [( yz ) y ]. The class of Bol loops includes all Moufang loops and all Bruck loops. For a commutative quasigroup of odd order, the identity mapping is a complete mapping and the same result is true for Bol loops of odd order.
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