I can't wrap my head around this area in mathematics. What is a group, a, semigroup, what is a field, a ring, an abelian group?
I read all sorts of texts, but it's so abstract. I can't solve problems in this field of mathematics because I don't understand the main idea behind this. Please, can someone explain it in simple English?
What do the properties of relations like reflexivity, symmetry, antisymmetry and transitivity mean? Do you have any examples or explanations?
In general, I try to explain abstract algebra as the study of common operations. For example, we use the "+" operation in many contexts: we can add integers, real numbers, complex numbers, vectors, polynomials, matrices, continuous functions, hours in the day, and in many other contexts.
All of these "+"'s have similar properties, they are commutative and associative, adding two objects of the same type results in another object of the same type, they have a zero (an identity), and they have negatives.
What abstract algebra may be thought of is the study of all of these "+"'s at the same time. You are trying to separate the properties that arise from the particular objects being added from the way that "+" acts.
For example, just from the properties of "+", one can determine that the zero (the identity) is unique (there's only one object that acts like a zero). It's not something special about the real numbers or continuous functions, it's something that must happen because of the way that addition works.