This is a story about what external algebra is, what it consists of. Surprisingly, there are almost no articles on external algebra on Habré, despite the fact that its applied value is no less than, for example, relational algebra. Outer algebra is a mixture of set theory, algebra, and combinatorics. It is the basis for understanding spaces, therefore, to one degree or another, it is present in almost all branches of mathematics. Despite the fact that her postulates are extremely simple.
Our presentation differs from the traditional one - we focus not so much on the accuracy of the wording as on the transmission of the essence. We deliberately use notations that differ from the generally accepted ones to simplify the wording. This article is to get the gist of it. Then it will be easier.
In the first part, we define the space based on the rules of the outer product and addition of objects. In the second, we add metric properties to the space. In general, we will go from the outer product to the representation of arbitrary graphs in the form of an algebraic expression. On the way, we will get acquainted with the basic ideas and tools of external algebra.
The essence of external algebra is that objects can be multiplied. If you are not impressed by this, then again - you can multiply any objects, not just numbers. You can multiply people, cities, computers, accounts and everything else that fits the concept of an object. This product of objects is called "external multiplication". It is a historically developed term to distinguish the outer product from the inner product, which is usually used to denote a dot product.
Usually, the outer product of mathematics is denoted by a wedge. But this sign is inconvenient. Firstly, it is not on the keyboard. Secondly, there is no other multiplication other than external multiplication for arbitrary objects. Therefore, we will denote the outer product with an asterisk. Here it is, element-by-element multiplication: .
Looks familiar, right? But unlike the usual product of numbers, the outer product is anticommutative - in other words, it changes sign when the factors are rearranged. Like this:
This is an important property that is key to outer algebra. From it, in particular, it follows that the external product of an element by itself can only be zero - an unsigned object:.
What is the meaning of the "outer work". In simple terms, it is nothing more than combining objects into a list. Therefore, when there are many elements, it is convenient to designate them with a list. We mark such an ordered list with square brackets:. It should be remembered that the list has a sign - when rearranging adjacent elements, the list sign changes to the opposite:. As a consequence, such a list cannot contain the same elements.