 If we want to be able to solve equations, it helps for our algebraic structure to have some further structure. Let's see what we might want, and while we're at it, let's also name things as we go. So let's think about that. If G star is an algebraic structure, then A and B elements of G means that A star B is defined. It's equal to something. But sometimes it's not in G. For example, in N minus, where our operands are the set of natural numbers and our operation is subtraction, some subtractions are not natural numbers. For example, 3 minus 8. And so this suggests a useful property. Let G be a set and star a binary operation. We say that G is closed under star if, given any A and B in G, we have A star B also an element of G. And since not all algebraic structures have this useful property, let's go ahead and give it a name. We'll define a magma, an algebraic structure G star, where G is closed under star. So sets like the set of natural numbers or the set of real numbers have an infinite number of terms, but we can also consider sets with a finite number of terms. In that case, we can represent the result of A star B using a Cayley table. These are named after Arthur Cayley, a 19th century British mathematician, who played a major role in the development of what we now call abstract algebra. A Cayley table is like a multiplication table where the first operand A provides the row labels, the second operand B provides the column labels, and the product A star B are the table entries. Now, fair warning, in Cayley's original description, the first operand was actually the column labels, and the second operand was the row labels. And some authors will use their Cayley tables as Cayley wrote them. So whenever you see a Cayley table from a new author, it's important to identify which of the two conventions they're using. We'll use this more because it's more consistent with what we're used to seeing in linear algebra and the description of matrices. For example, let G be the set of three elements A, B, and C, and suppose our binary operation star has the Cayley table. Let's use this to find A star B. And so to find A star B, we go to the row with label A, then go to the column with label B. And the entry in the A, row, and B column is A star B, and so we write down that A star B is equal to C. Now our goal is to build up an algebraic structure that can be used to solve equations. If we have a single operation of the Cayley table, we can look up the answer. Or can we? Well, let's try it out. Suppose we have our set of three elements, and suppose we have a Cayley table that looks like this, and let's try to solve a few equations. For example, let's solve for X, B star X equal to B, and let's also solve for Y. Y star C is equal to B. So if we want to solve B star X equal to B, we want B times something to give us B. And so we go to the B row of the table, and we look across and we see that B star A is equal to B. And so that tells us that X is equal to A. But wait, there's more. We also see that B star B is equal to B, and so X could also be equal to B. Similarly, if I want to solve the equation Y star C equal to B, C is my second operand, so I'll go to the C column of the table, and that tells us that A star C is equal to B, and so Y is equal to A. And we also see that C star C is equal to B, and so Y is equal to C. Now while school algebra does have equations with more than one solution, for example, X squared equals 25, we're used to the idea that the simplest equations have unique solutions. In order for this to happen, it's important that the entries in our Cayley table satisfy the following. Every entry in a column is different, and every entry in a row is different. And this is important enough that we give it a specific name. This is the Latin square property. And so we say that a Cayley table has the Latin square property if every entry appears exactly once in each row and once in each column. And this is a new requirement, and so this gives us a new algebraic structure. A quasi-group is an algebraic structure G star, where G is closed under star, and the Cayley table for star satisfies the Latin square property. So hey, I got this great idea for a game. Suppose I give you a Cayley table, and I tell you that it's a quasi-group. From some of the Cayley table, you should be able to reproduce all of it. So remember, definitions are the whole of mathematics, all else is commentary. We know that this is a quasi-group, so let's pull in our definition of a quasi-group. The definition of a quasi-group talks about the Latin square property, so let's pull in the Latin square property. And so we know that every entry appears exactly once in each row and once in each column. Remember, the entries in the top row and the left column are actually the factors, so they don't count in this once per row, once per column requirement. So I have an entry A here, and so these remaining entries, either the entries in the same row or the entries in the same column, can't be A. They have to be something else. So let's reason our way through this. The first column already contains an A, so the remaining entries must be B and C. But since the second row already contains a B, then the entry in the second row, first column, must be C, and that means the last entry must be B. Now let's take a look at that second column. The second column already contains a B, so again the remaining entries have to be A and C. But since the first row already contains an A, the entry in the first row, second column, must be C, and the entry in the last row must be A. And finally, let's take a look at the entries in the last column. In the first row, we already have A and C, that means this last entry has to be B. Second row, we already have C and B, so the last entry has to be A. And then third row, we already have a B and an A, which means the last entry has to be C.