If you’re looking at groups, you’re looking at an abstraction of the idea of numbers, to try to reduce it to minimal properties. As I’ve already explained, a group is a set of values with one operation, and which satisfies several simple properties. From that simple structure comes the
\nbasic mathematical concept of symmetry.<\/p>\n
Once you understand some of the basics of groups and symmetry, you can move in two directions. You can ask “What happens if I add<\/em> something?”; or you can ask “What happens if I remove<\/em> something?”. <\/p>\n You can either add operations – which can lead you to a two-operation structure called a ring; or you can add properties – in which the simplest step leads you to something called an abelian group<\/em>. When it comes to removing, you can remove properties, which leads you to a simpler structure called a groupoid<\/em>. Eventually, I’m going to follow both the upward and the downward paths. For now, we’ll start with the upward path, since it’s easier.<\/p>\n Building up from groups, we can progress to rings<\/em>. A group captures one simple property of <\/p>\n A ring is a set of values R, along with two operations, “+” and “×”. We’ll generally call the The first operation in the ring is addition. In the ring, (R,+) is an abelian group, with identity The second operation, multiplication, in a ring doesn’t have to have all of the group operation Finally, the relationship between the two operations has to satisfy one condition: the addition So with all of this stuff, what have we really described? We’ve created something that is an So, obviously, the integers with integer addition and multiplication are a ring. So are the real numbers, with real addition and multiplication, and the complex numbers with complex addition and multiplication. Even single-variable polynomials form a ring using simple polynomial addition and multiplication.<\/p>\n More interestingly, you can form fields from some very different things, which There are also a ton of example rings that are easier to understand in a category theoretic If you’re looking at groups, you’re looking at an abstraction of the idea of numbers, to try to reduce it to minimal properties. As I’ve already explained, a group is a set of values with one operation, and which satisfies several simple properties. From that simple structure comes the basic mathematical concept of symmetry. Once […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[26],"tags":[],"class_list":["post-570","post","type-post","status-publish","format-standard","hentry","category-group-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-9c","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/570","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=570"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/570\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=570"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=570"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=570"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\na set of number-like objects. A ring brings us closer to capturing the structure of the system
\nof numbers. The way that it does this is by adding a second operation. A group has one operation
\nwith symmetric properties; a ring adds a second symmetric operation, with a well-defined relationship
\nbetween the two operations.<\/p>\n
\ntwo operations addition and multiplication, although it’s important to remember that they are
\nnot<\/em> the standard numeric add and multiply that we’re familiar with: these are abstract
\noperations, and we can construct rings in which the “add” operation has very little similarity with what
\nwe think of as addition.<\/p>\n
\nvalue is written “0”, and where the inverse of value a∈R with respect to “+” is written “-a”.<\/p>\n
\nproperties. (Remember, what we’re trying to do is look at very simple structures, so we don’t want to
\nadd more requirements that we absolutely have to; the more abstract we can leave the definition, the
\nmore structures the definition can encompass. We want to add just enough to be able to explore the
\nconcepts.) The only requirements on the second operation are that it is associative, and has an identity
\nelement. There’s no requirement that it be commutative, and there’s no requirement that there be
\nmultiplicative inverses.<\/p>\n
\noperation must be distributive over the multiplication operation. To be formal, ∀a,b,c∈R,
\na×(b+c) = a×b + a×c ∧ (a+b)×c = a×c + b×c.<\/p>\n
\nabstract definition of something that is very similar to the integers. Integers with addition form a
\ngroup. But integers can’t form a group with multiplication, because there are multiplicative inverses in
\nthe integers. So we’ve created a structure that lets us describe things that behave like integers. <\/p>\n
\nturn out to share this basic structure. For example, if you take an arbitrary
\nset S, then the powerset of S (that is, the set of all subsets of S) form
\na ring with symmetric set difference as the addition operation, and intersection as the
\nmultiplication operation. This one always amazes me, from the first time I saw it. When
\nyou study groups and rings, you tend to see a lot of examples that all, in some sense, satisfy
\nyour intuition for either addition or multiplication. Then you encounter this – and it really
\nhits home that this abstraction process has given you something that’s really quite different – these basic properties of addition and multiplication on the integers can describe very different
\nthings that don’t fit your intuition well at all.<\/p>\n
\nmodel. or example, if you have an Abelian group, (G,+), then the set of endomorphisms
\nof (G,+) are a ring are addition and composition of the endomorphisms, using the
\ncategory-theoretic version of morphism addition. I won’t go into that in any more detail
\nhere, because I’m eventually going to get into looking at a lot of this abstract algebra
\nmaterial in terms of category theory, so I’ll save it for then.<\/p>\n","protected":false},"excerpt":{"rendered":"