Lectures on Universal Algebra

1 Algebras In this first section we will consider some common features of familiar algebraic structures such as groups, rings, lattices, and boolean algebras to arrive at a definition of a general algebraic structure. Recall that a group G consists of a nonempty set G, along with a binary operation · : G → G, a unary operation −1 : G → G, and a constant 1 G such that • x · (y · z) = (x · y) · z for all x, y, z ∈ G, A group is abelian if it additionally satisfies: x · y = y · x for all x, y ∈ G. A ring R is a nonempty set R along with binary operations +, ·, a unary operation −, and constants 0 R and 1 R which satisfy • R, along with +, −, and 0 R is an abelian group. • x · (y · z) = (x · y) · z for all x, y, z ∈ G. • x · (y + z) = (x · y) + (x · z) and (y + z) · x = (y · x) + (z · x) for all x, y, z ∈ G.