different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. See small Latin squares and quasigroups. Infinite quasigroups For a countably infinite quasigroup Q, it is possible to imagine an infinite array in which every row and every column corresponds to some element q of Q, and where the element is in the row corresponding to a and the column responding to b. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once. For an uncountably infinite quasigroup, such as the group of non-zero real numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in a sequence. (This is somewhat misleading however, as the reals can be written in a sequence of length , assuming the well-ordering theorem.) Inverse properties The binary operation of a quasigroup is invertible in the sense that both and , the left and right multiplication operators, are bijective, and hence invertible. Every loop element has a unique left and right inverse given by A loop is said to have (two-sided) inverses if for all x. In this case the inverse element is usually denoted by . There are some stronger notions of inverses in loops which are often useful: A loop has the left inverse property if for all and . Equivalently, or . A loop has the right inverse property if for all and . Equivalently, or . A loop has the antiautomorphic inverse property if or, equivalently, if . A loop has the weak inverse property when if and only if . This may be stated in terms of inverses via or equivalently . A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop which satisfies any two of the