Reddit mentions of Algebra (AMS Chelsea Publishing)

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

I have to second Dummit and Foote as a supplement to Lang's text, they're pretty much complete opposites; where Lang is very to the point (terse, some may say) and from a very abstract viewpoint, Dummit and Foote has a lot of exposition and examples and is done from, what at least what I would call, an appropriate level for a first graduate course in abstract algebra. It also has an appendix that deals with category theory, it's nothing extensive but it may help you become more familiar with the ideas of category theory. I am currently using this book for a graduate course in algebra so I have some familiarity with it; it is a bit too wordy for my tastes but that may be your thing.

A book with which I have limited experience but quite like so far is Mac Lane and Birkhoff's Algebra it's done with the same general perspective as Dummit and Foote but it has a bit more category theory (it is introduced at the end of the third chapter and the entire fifteenth chapter is dedicated to category theory), it isn't terse but it is less wordy than Dummit and Foote.

Another (very) popular choice (but one with which I have no experience) is Aluffi's Algebra: Chapter 0 it develops category theory pretty much from the start and supposedly is much less terse than Lang (I only say supposedly as I have no first hand experience with it).

If you want something that only deals with category theory, the classic text is Mac Lane's Category Theory for the Working Mathematician I have found looking at this book for a long period of time has helped me with understanding/getting used to categorical ideas. I also have experience with this book for which you can find on the internet (legally) for free and I find it rather good.

Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.

The best book for an introduction I have read is:

Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&qid=1453926037&sr=8-1&keywords=algebra+maclane)

For more advanced stuff, and to secure the understanding better I recommend this book:

Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&qid=1453926180&sr=8-1&keywords=topoi)

Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.

For type theory and lambda calculus I have found the following book to be the best:

Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&qid=1453926270&sr=8-2&keywords=type+theory)

The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.

I dunno about “undergraduate”, but you could try Birkhoff & Mac Lane or Greub. Those are both kind of old, so someone else may have a better idea.

Mac Lane and Birkhoff is my favorite math text I've ever read, but the things you have listed as "further topics" in your second semester are mostly absent unfortunately.

I would advise you not to start with category theory, but abstract algebra. Mac Lane and Birkhoff's book Algebra is excellent and well worth the money in hardback. It covers things like monoids, groups, rings, modules and vector spaces, all of which are -- not coincidentally -- typical examples of structures that form categories. Saunders Mac Lane invented category theory along with Samuel Eilenberg, and Birkhoff basically founded universal algebra, so you cannot find a more authoritative text.

Edit: The other thing that will really help you is a basic understanding of preorders and posets. I don't have a book that deals exclusively with this topic, but any introduction to lattice theory, logical semantics or denotational semantics of programming languages will treat it. I would recommend Paul Taylor's Practical Foundations of Mathematics, though the price on Amazon is very steep. You can look through it here: http://paultaylor.eu/~pt/prafm/

If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):

Basic Algebra by Jacobson

Algebra by Lang

Algebra by MacLane/Birkhoff

Algebra by Herstein

Algebra by Artin

etc

But there are other books that are "essential" to modern readers:

Chapter 0 by Aluffi

Basic Algebra by Knapp

Algebra by Dummit/Foot

For basic Algebra(Linear, Multilinear bla, bla, bla) there exists an amazing book called "Algebra" by Saunders Maclane and Garett Birkhoff

I don't know what second/third semester Calculus means. Is it proof-based or non-proof based? Is it a regular Calculus sequence or is it Analysis?

Birkhoff and Mac Lane's Algebra goes a long way, and gets you used to Mac Lane's style.

Algebra by Saunders MacLane and Garret Birkhoff is the best algebra book I have ever encountered.

For algebra, I'd recommend Mac Lane/Birkhoff. They may not be as comprehensive as some other texts but to me, they are more motivating, and will probably provide a better introduction to categorical thinking.

For linear algebra, I'm going to suggest something slightly unusual: Kreyszig's Introductory Functional analysis with applications. Functional analysis is essentially linear algebra on infinite dimensional spaces, and it generalizes a lot of the results in finite dimensions. Kreyszig does a good job motivating the reader. I can definitely sit down and read it for hours, much longer than I can for other books, and I definitely don't consider myself an analyst. However, it could be difficult if you are not familiar with basic topology and never seen linear algebra before.