I am busy doing an undergraduate course called "Fundamentals of Mathematics". It is not well-defined as there is no syllabus nor recommended textbook (there are lectures and notes), but the course introduces one to quite complicated theoretical aspects of mathematics. I would like to find a textbook for this course which would help me to understand what is being taught.
What we have already covered or are covering:
Magmas and unitary magmas, preorders and orders, semilattices and lattices, semigroups, monoids, closure operators, equivalence relations (I am familiar with the definition of an equivalence relation, but I can revise the place a quotient set has in a canonical factorization of a map and so on), Russell Paradox, Boolean algebra, cardinality, categories, morphisms (including monomorphisms, endomorphisms and isomorphisms), sub-algebras, set-theoretic definition of natural numbers.
What I predict we will still cover, based on the notes:
Canonical factorization of homomorphisms, quotient algebras, classical algebraic structures, quotient groups, rings and modules, semirings and semimodules, pointed categories, products and coproducts, direct sums, free algebras and free semimodules.
I already have a textbook called "Introduction to Abstract Algebra", but this focuses on, and goes into depth about, algebraic structures, as opposed to briefly constructing the many structures mentioned above.
Try books by Serge Lang, Spivak, and in that direction, I mean I'm throwing names out there because I have these books and the amount of sheer detail is staggering, this a start, branch out from here, locate common publishers and cross-check user reviews